UBC Theses and Dissertations

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UBC Theses and Dissertations

A comparison of numerical algorithms for determining electrical resistance and reactance using subdivision… De Arizon, Paloma 1984

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A COMPARISON OF NUMERICAL ALGORITHMS FOR DETERMINING ELECTRICAL RESISTANCE AND REACTANCE USING SUBDIVISION OF THE CABLE CONDUCTORS by PALOMA DE ARIZON E l e c . E ng.(Hons.), U n i v e r s i d a d Simon B o l i v a r , V e n e z u e l a , 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l E n g i n e e r i n g ) We ac c e p t t h i s t h e s i s as c o n f o r m i n g . to the r e q u i r e d standard* THE UNIVERSITY OF BRITISH COLUMBIA * o b September, 1984 > *" ( c ) Paloma de A r i z o n , 1984 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f E l e c t r i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 4, 1984 DE-6 (3/81) - i i -A B S T R A C T I n t h i s w o r k , t h e i n d u c t a n c e a n d r e s i s t a n c e o f d i f f e r e n t c a b l e c o n f i g u r a t i o n s a r e o b t a i n e d b y d i v i d i n g t h e c o n d u c t o r s i n t o s m a l l e r f i l a m e n t s o f a s p e c i f i c c r o s s - s e c t i o n a l s h a p e . D i f f e r e n t f i l a m e n t s h a p e s a r e s t u d i e d , a n d t h e a d v a n t a g e s a n d d i s a d v a n t a g e s o f e a c h o n e a r e c o n s i d e r e d . T h e i m p e d a n c e o f c a b l e s i s a n i m p o r t a n t p a r a m e t e r f o r m o s t s t u d i e s i n v o l v i n g c a b l e s y s t e m s . Y e t t h e r e a r e s o m e t y p e s o f c a b l e s , s u c h a s p i p e t y p e c a b l e s , w h e r e o n l y e x p e r i m e n t a l r e s u l t s a r e a v a i l a b l e , a n d w h e r e a n a l y t i c a l r e s u l t s a r e n o t a l w a y s a c c u r a t e e n o u g h b e c a u s e t h e y d o n o t t a k e s k i n a n d p r o x i m i t y e f f e c t s i n t o a c c o u n t . W i t h t h e t e c h n i q u e d e s c r i b e d h e r e , t h e p r o b l e m o f c a b l e p a r a m e t e r c a l c u l a t i o n h a s b e e n s o l v e d . I t i s b a s e d o n t h e f o r m u l a t i o n o f a [ Z ] m a t r i x , w h i c h c a n t a k e e a r t h r e t u r n e f f e c t s i n t o a c c o u n t . T h e e a r t h r e t u r n i m p e d a n c e i t s e l f i s i n c l u d e d w i t h a n a l y t i c a l f o r m u l a e , i n o r d e r t o s p e e d u p t h e c a l c u l a t i o n s . T h e i m p e d a n c e o f c a b l e s w i t h n o n m a g n e t i c m a t e r i a l s c a n b e c a l c u l a t e d w i t h a h i g h d e g r e e o f a c c u r a c y . F o r m a g n e t i c m a t e r i a l w i t h s a t u r a t i o n e f f e c t s , s o m e r e a s o n a b l e a p p r o x i m a t i o n s a r e s m a d e . T h e i m p e d a n c e o f s t r a n d e d c o n d u c t o r s c a n b e c a l c u l a t e d a s w e l l , n e g l e c t i n g t h e e f f e c t o f s p i r a l l i n g . T h e c u r r e n t d i s t r i b u t i o n i n a p i p e t y p e c a b l e i s u s e d a s a n e x a m p l e t o d e m o n s t r a t e t h a t s k i n a n d p r o x i m i t y e f f e c t s a r e t a k e n i n t o a c c o u n t w i t h t h i s t e c h n i q u e . - i i i -TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i LIST OF SYMBOLS v i i i ACKNOWLEDGEMENTS x 1. INTRODUCTION 1 1.1 A B r i e f Review of Methods f o r the C a l c u l a t i o n of Cable Impedance. 1 1.2 S k i n and P r o x i m i t y E f f e c t . 3 1.2.1 S k i n E f f e c t . 4 1.2.2 P r o x i m i t y E f f e c t . 7 1.3 A B r i e f E x p l a n a t i o n of the Method of S u b c o n d u c t o r s . 7 2. THEORY AND FORMATION OF THE IMPEDANCE MATRIX 10 2.1 G e n e r a l Assumptions. 10 2.2 E q u a t i o n s . 11 2.2.1 C i r c u l a r F i l a m e n t s . 12 2.2.2 Square F i l a m e n t s . 17 2.2.3 E l e m e n t a l s . 20 3. REDUCTION OF THE LARGE IMPEDANCE MATRIX 28 3.1 R e d u c t i o n of the Large Impedance M a t r i x . 29 4. RETURN PATH IMPEDANCE 33 - i v-4.1 A n a l y t i c a l Ground Return Formulae. 35 4.2 Representing the ground as One Undivided Conductor. 38 4.2.1 Mutual Impedance between a subconductor and ground with common r e t u r n i n another subconductor. 40 5. PIPE TYPE CABLES 42 6. STRANDED CONDUCTOR 46 7. RESULTS 53 7.1 Comparison of the Method of S u b d i v i s i o n with Standard methods. 53 7.2 Comparison of Ground Return Formulae. 61 7.3 Pipe Type Cables. 70 7.3.1 V a r i a t i o n of the [Z] matrix with p e r m e a b i l i t y of p i p e . 71 7.3.2 V a r i a t i o n of the [Z] matrix with frequency. 73 7.3.3 Pipe Type Cable C o n f i g u r a t i o n s . 75 7.3.4 Current D i s t r i b u t i o n i n Pipe Type Cables. 78 7.4 Stranded Conductors. 85 8. CONCLUSIONS 88 LIST OF REFERENCES 90 APPENDIX A 93 A . l F i n i t e D i f f e r e n c e S o l u t i o n . 93 A.2 F i n i t e Element S o l u t i o n . 98 A.3 Comparison with S u b d i v i s i o n Method 100 - V -LIST OF TABLES TABLE 7.1 V a r i a t i o n o f R e s i s t a n c e w i t h F r e q u e n c y f o r t h e Four D i f f e r e n t M e t hods. 55 7.2 V a r i a t i o n o f I n d u c t a n c e w i t h F r e q u e n c y f o r t h e Four D i f f e r e n t Methods. 55 7.3 V a r i a t i o n o f Impedance o f a C o a x i a l C a b l e w i t h P o s i t i o n o f t h e C e n t r a l C o n d u c t o r . 59 7.4 V a r i a t i o n o f t h e Impedance o f a B u r i e d C o n d u c t o r w i t h Depth o f B u r i a l . 61 7.5 S e l f Impedance o f t h e C e n t r a l C o n d u c t o r o f a C o a x i a l C a b l e w i t h Ground R e t u r n . 64 7.6 S e l f Impedance o f t h e S h e a t h o f a C o a x i a l C a b l e w i t h Ground R e t u r n . 65 7.7 M u t u a l Impedance between S h e a t h and C e n t r a l C o n d u c t o r i n P r e s e n c e o f E a r t h . 66 7.8 Change i n P i p e P a r a m e t e r s w i t h P e r m e a b i l i t y o f t h e P i p e . 73 7.9 F r e q u e n c y V a r i a t i o n o f P i p e Type C a b l e M a t r i x . 74 7.10 V a r i a t i o n o f Impedance w i t h C o n f i g u r a t i o n o f I n t e r n a l C o n d u c t o r s . 82 7.11 V a r i a t i o n w i t h F r e q u e n c y o f t h e R e s i s t a n c e o f a One L a y e r S t r a n d e d C o n d u c t o r . 85 7.12 V a r i a t i o n w i t h F r e q u e n c y o f t h e R e s i s t a n c e o f a Two L a y e r s S t r a n d e d C o n d u c t o r 86 -v i -LIST OF ILUSTRATIONS FIGURE 1.1 Conductor s u b d i v i s i o n i n t o "g" f i l a m e n t s of equal cross s e c t i o n a l area and f i l a m e n t r e s i s t a n c e "R^". 6 2.1 S u b d i v i s i o n of the main conductor with the three types of f i l a m e n t s . 10 2.2 Loops formed by two subconductors with a common r e t u r n . 12 2.3 A lumped constant parameter model of an N conductor t r a n s m i s s i o n l i n e . 13 2.4 Geometry of subconductors. 14 2.5 Rectangular loops - Neumann Formulas. 19 2.6 "Elemental" c r o s s - s e c t i o n a l shapes. 21 2.7 Filaments - "elemental" impedance c a l c u l a t i o n . 22 2.8 P a r a l l e l m o n o l i t h i c f i l a m e n t s . 24 2.9 "Elementals" r e l a t i v e l y f a r appart. 25 2.10 "Elementals" which are c l o s e t o g e t h e r . 26 2.11 A p a i r of p a r a l l e l s t r a i g h t l i n e s . 27 4.1 S u b d i v i s i o n of the s o i l . 35 4.2 Two conductors with common ground r e t u r n . 40 4.3 C i r c u i t of one conductor and the ground with common r e t u r n i n another conductor. 40 5.1 V a r i a t i o n of r e l a t i v e p e r m e a b i l i t y with c u r r e n t through the pipe. 43 5.2 Phasor diagram f o r one conductor of a three core c a b l e . 45 6.1 Stranded conductor. 48 7.1 C o a x i a l c a b l e . 53 7.2 V a r i a t i o n of Resistance with Frequency f o r the Four D i f f e r e n t s Methods. 56 - v i i -7.3 V a r i a t i o n of Inductance with Frequency f o r the Four D i f f e r e n t Methods. 56 7.4 V a r i a t i o n of the impedance with the p o s i t i o n of the c e n t r a l conductor. 59 7.5 V a r i a t i o n of the impedance of a b u r i e d conductor with depth of b u r i a l . 62 7.6 V a r i a t i o n with Frequency of Self-Impedance of C e n t r a l Conductor i n a C o a x i a l Cable. 67 7.7 V a r i a t i o n with Frequency of S e l f Impedance of Sheath i n a C o a x i a l Cable. 68 7.8 V a r i a t i o n with Frequency of Mutual Impedance Between Sheath and C e n t r a l Conductor. 69 7.9 Pipe type c a b l e . 72 7.10 Cradle c o n f i g u r a t i o n . 77 7.11 Close t r i a n g u l a r c o n f i g u r a t i o n . 77 7.12 Open t r i a n g u l a r c o n f i g u r a t i o n . 77 7.13 S u b d i v i s i o n of Pipe Type Cable. 79 7.14 One conductor i n center of p i p e . 80 7.15 One conductor i n e c c e n t r i c p o s i t i o n . 80 7.16 Three conductors i n pipe. 80 7.17 Current D i s t r i b u t i o n i n the P i p e . Cases a) and b ) . 83 7.18 Current D i s t r i b u t i o n i n the I n t e r n a l Conductor. Cases a) and b ) . 83 7.19 Current D i s t r i b u t i o n i n the P i p e . Cases b) and c ) . 84 7.20 Current D i s t r i b u t i o n i n I n t e r n a l Conductor. Cases b) and c ) . 84 7.21 Stranded conductor. 87 a . l S u b d i v i s i o n of a two dimensional continuum. 94 a.2 T y p i c a l g r i d p o i n t 0, i t s surrounding meshes and g r i d p o i n t s . 95 - v i i i -LIST OF SYMBOLS B f l u x d e n s i t y D., d i s t a n c e between c o n d u c t o r s 1 and q l q e 2.718281828 f f r e q u e n c y GMR Geometric mean r a d i u s i , I c u r r e n t j complex o p e r a t o r / -1 km k i l o m e t r e s m / j tou/p m metres q s u b s c r i p t d e n o t i n g r e t u r n p ath r,R r a d i u s , r e s i s t a n c e v,V v o l t a g e X r e a c t a n c e Z impedance K ,K, B e s s e l f u n c t i o n s o' 1 l o g common l o g a r i t h m In n a t u r a l l o g a r i t h m Hz h e r t z U Q a b s o l u t e p e r m e a b i l i t y of f r e e space = 4^ 10~^ H/m u = u 0 ^ r p e r m e a b i l i t y u r r e l a t i v e p e r m e a b i l i t y TT 3.1415926 Y 0.5772157 E u l e r s c o n s t a n t £2 ohm - i x -W 2 Tt f i|) f l u x l i n k a g e p r e s i s t i v i t y Q m S, c a b l e l e n g t h m i n d u c t a n c e H e n r y n n u m b e r o f o u t e r s t r a n d s A m a g n e t i c p o t e n t i a l v e c t o r B = V x A v r e l u c t i v i t y 1 / u 2 J c u r r e n t d e n s i t y A m p / m -x-ACKNOWLEDGEMENTS I w o u l d l i k e t o e x p r e s s my t h a n k s t o a l l t h o s e p e r s o n s t h a t h e l p e d made my s t u d i e s a t UBC a v e r y e n j o y a b l e a n d p r o f i t a b l e e x p e r i e n c e and i n s p e c i a l : t o Dr. H.W. Dommel, f o r h i s i n v a l u a b l e h e l p and d i r e c t i o n t h r o u g h o u t t h i s p r o j e c t . t o S r i v a l l i p u r a n a n d a n , f o r h i s h e l p i n r u n n i n g many of t h e t e s t c a s e s u s e d t o v a l i d a t e t h e m o d e l p r e s e n t e d i n t h i s t h e s i s . t o my o f f i c e mates L u i s and S u b r o t o , f o r t h e i r c o n t i n u o u s s u p p o r t and encouragement t o my husband N e l s o n , n o t o n l y f o r h i s c o n s t a n t l o v e and s u p p o r t , but a l s o f o r h i s h e l p and s u g g e s t i o n s throughout t h i s p r o j e c t . t o my p a r e n t s and b r o t h e r s , f o r t h e i r l o v e and e m o t i o n a l s u p p o r t . - 1 -1.INTRODUCTION 1.1 A B r i e f R e v i e w o f M e t h o d s f o r t h e C a l c u l a t i o n o f C a b l e  Impedances I n r e c e n t y e a r s the c o s t a s s o c i a t e d w i t h r i g h t s of way of t r a n s m i s s i o n l i n e s has become an i n c r e a s i n g l y i m p o r t a n t f a c t o r i n t h e o v e r a l l d e s i g n o f power s y s t e m s , e s p e c i a l l y i n u r b a n a r e a s . T h e r e f o r e , compact a r r a n g e m e n t s s u c h a s i n p i p e t y p e c a b l e s and i n s u b s t a t i o n s w i t h g a s i n s u l a t e d b u s e s h a v e become more i m p o r t a n t . F o r most o f t h e s t u d i e s i n v o l v i n g o v e r h e a d t r a n s m i s s i o n l i n e s and c a b l e s y s t e m s a l i k e , s u c h as f a u l t s t u d i e s and s u r g e p r o p a g a t i o n s t u d i e s , r e a s o n a b l y a c c u r a t e i m p e d a n c e d a t a i s needed. T h i s work i s c e n t e r e d on the s t u d y of c a b l e p a r a m e t e r s , a l t h o u g h t h e t e c h n i q u e p r e s e n t e d can e a s i l y be a p p l i e d t o t h e a n a l y s i s of overhead t r a n s m i s s i o n l i n e s as w e l l U n d e r g r o u n d c a b l e s y s t e m s h a v e b e e n s t u d i e d by many a u t h o r s , w i t h t e c h n i q u e s w h i c h can be d i v i d e d i n t o t h r e e major c a t e g o r i e s : a ) A n a l y t i c a l T e c h n i q u e s , i n w h i c h a t t e m p t s a r e made t o e x p r e s s t h e s o l u t i o n s o f t h e a p p r o p r i a t e f i e l d e q u a t i o n s i n t e r m s of f u n c t i o n s s u c h as B e s s e l f u n c t i o n s . Here i t i s p o s s i b l e to i n c l u d e t h e work - 2 -o f S i m m o n s [ 2 4 ] , w h i c h r e s u l t e d i n t h e p u b l i c a t i o n o f s t a n d a r d c h a r t s f o u n d i n many h a n d b o o k s . F o r s i n g l e - c o r e d ( c o a x i a l ) c a b l e s t h e r e s u l t s o b t a i n e d by S c h e l k u n o f f [ 2 8 ] a r e w e l l known. C a r s o n [ 6 ] , P o l l a c z e k [ 9 ] , Wedepohl and W i l c o x [ 3 ] have a l s o d e r i v e d e q u a t i o n s f o r t h e impedance o f underground c a b l e s w i t h ground r e t u r n . Smith and B a r g e r [ 2 5 ] , L e w i s e t a l [ 2 6 ] and o t h e r s have c a l c u l a t e d t h e i m p e d a n c e o f c o n c e n t r i c n e u t r a l c a b l e s used i n d i s t r i b u t i o n systems. I t i s i m p o r t a n t a t t h i s p o i n t t o emphasize t h a t t h e c a b l e c o n f i g u r a t i o n s s o l v e d by t h e s e a n a l y t i c a l f o r m u l a e a r e m a i n l y s y m m e t r i c a l c o n f i g u r a t i o n s and t h a t i n most o f t h e f o r m u l a e u s e d i n i m p e d a n c e c a l c u l a t i o n s , f a c t o r s a r e r e q u i r e d to c o r r e c t f o r s k i n and p r o x i m i t y e f f e c t s , b ) E x p e r i m e n t a l d a t a : As p o i n t e d out a b o v e , c e r t a i n n o n - s y m m e t r i c a l c o n f i g u r a t i o n s s u c h as p i p e t y p e c a b l e s , have n o t y e t been s o l v e d w i t h a n a l y t i c a l f o r m u l a e , due t o the c o m p l i c a t e d c o n f i g u r a t i o n o f t h e e l e c t r o m a g n e t i c f i e l d s . T h e r e f o r e e x p e r i m e n t s w e r e done t o o b t a i n t h e i m p e d a n c e o f p i p e t y p e c a b l e s . Here the e x p e r i m e n t s done by Neher [ 7 ] , Thomas [8] and Kershaw and most r e c e n t l y the one done by K a t z et a l [21] must be m e n t i o n e d . U n f o r t u n a t e l y ' m o s t o f t h e s e - 3 -e x p e r i m e n t s have been l i m i t e d t o power f r e q u e n c y (60 H z ) , and t h e r e s u l t s do n o t t h e r e f o r e p r o v i d e s u f f i c i e n t d a ta f o r t r a n s i e n t s t u d i e s . c ) N u m e r i c a l T e c h n i q u e s , w i t h two major a p p r o a c h e s : W i t h t h e f i r s t a p p r o a c h t h e s o l u t i o n o f t h e f i e l d e q u a t i o n s i s o b t a i n e d by f i n i t e e l e m e n t o r f i n i t e d i f f e r e n c e m e t h o d s W e i s s a n d C s e n d e s [ 1 8 ] , S i l v e s t e r [ 1 9 ] ( s e e A p p e n d i x ) . W i t h t h e s e c o n d a p p r o a c h , t h e impedance o f c a b l e s i s o b t a i n e d by d i v i d i n g a l l t h e c o n d u c t o r s ( i n c l u d i n g g r o u n d ) i n t o s m a l l e r s u b c o n d u c t o r s o f s p e c i f i c s h a p e . T h e s e n u m e r i c a l t e c h n i q u e s t a k e s k i n a n d p r o x i m i t y e f f e c t s i n t o a c c o u n t , as w e l l as o t h e r m a j o r e f f e c t s s u c h as eddy c u r r e n t s i n d u c e d i n s h e a t h s . The s e c o n d a p p r o a c h h a s b e e n u s e d by C o m e l l i n i e t a l [ 2 ] , Lucas and T a l u k d a r [ 4 ] , Graneau [ 1 ] and i s a l s o t h e a p p r o a c h used i n t h i s t h e s i s . C a b l e s w i t h s e c t o r - s h a p e d c o n d u c t o r s o r any i r r e g u l a r c r o s s s e c t i o n , can be e a s i l y s t u d i e d w i t h t h i s method. The Appendix d i s c u s s e s t h e r e a s o n why t h i s method was c h o s e n i n s t e a d of t h e t e c h n i q u e of f i n i t e e l e m e n t s . 1.2 S k i n and P r o x i m i t y E f f e c t s S e v e r a l phenomena r e l a t e d to t h e c u r r e n t d i s t r i b u t i o n i n c o n d u c t o r s w h i c h , f r o m a t h e o r e t i c a l s t a n d p o i n t , c a n a l l be s t u d i e d by M a x w e l l ' s e q u a t i o n s a r e c o v e r e d by t h e t e c h n i q u e used - 4 -i n t h i s t h e s i s T h e s e p h e n o m e n a i n c l u d e : - s k i n e f f e c t - p r o x i m i t y e f f e c t - e d d y c u r r e n t s i n c a b l e s h e a t h s - c u r r e n t t h r o u g h t h e s o i l - l o s s e s i n o v e r h e a d l i n e e a r t h w i r e s 1 T h e f i r s t t w o e f f e c t s a r e d i s c u s s e d i n d e t a i l . 1 . 2 . 1 S k i n E f f e c t I t i s w e l l k n o w n t h a t i n a c o n d u c t o r p l a c e d i n t o a n a l t e r n a t i n g m a g n e t i c f i e l d , t h e i n d u c e d c u r r e n t s w i l l t e n d t o c o n c e n t r a t e n e a r t h e s u r f a c e o f t h e c o n d u c t o r . T h i s s o - c a l l e d s k i n e f f e c t i s a l l t h e s t r o n g e r t h e h i g h e r t h e e l e c t r i c a l c o n d u c t i v i t y a n d t h e f r e q u e n c y o f t h e m a g n e t i c f i e l d . S k i n e f f e c t i s t h e p h e n o m e n a o f n o n u n i f o r m c u r r e n t d i s t r i b u t i o n o v e r t h e c r o s s s e c t i o n c a u s e d b y t h e t i m e v a r i a t i o n o f t h e c u r r e n t i n t h e c o n d u c t o r i t s e l f . a ) a d d i t i o n a l J o u l e h e a t i n g b ) a d e c r e a s e i n s e l f i n d u c t a n c e c ) c h a n g e s i n e l e c t r o m a g n e t i c f o r c e d i s t r i b u t i o n T h e d e c r e a s e i n t h e s e l f i n d u c t a n c e i s e a s y t o u n d e r s t a n d b e c a u s e t h e h i g h e r c u r r e n t d e n s i t y t o w a r d s t h e W h e n t h e m e t h o d i s a p p l i e d t o t h e s t u d y o f o v e r h e a d t r a n s m i s s i o n l i n e s . - 5 -c o n d u c t o r s u r f a c e r e d u c e s t h e s e l f l i n k a g e i n t h e i n n e r p a r t of t h e c o n d u c t o r s . T h e c u r r e n t d i s t r i b u t i o n f u r t h e r d e t e r m i n e s t h e e l e c t r o m a g n e t i c f o r c e s a c t i n g between t h e m e t a l l i c components of a c a b l e . The s t u d y of t h e s e f o r c e s i s one of t h e most d i f f i c u l t branches i n c a b l e d e s i g n . The u n i f o r m d i s t r i b u t i o n o f a g i v e n c u r r e n t o v e r t h e c r o s s s e c t i o n of a homogeneous c o n d u c t o r r e s u l t s i n t h e l o w e s t ohmic l o s s e s w h i c h t h i s amount o f t o t a l c u r r e n t can g e n e r a t e f o r a g i v e n r e s i s t i v i t y . Any d e p a r t u r e f r o m u n i f o r m c u r r e n t d i s t r i b u t i o n , as f o r example caused by s k i n e f f e c t , w i l l i n c r e a s e the l o s s e s and r e d u c e t h e c o n d u c t o r e f f i c i e n c y . T h e f a c t t h a t the l o s s e s a r e minimum f o r u n i f o r m c u r r e n t d i s t r i b u t i o n may be proven as f o l l o w s : L e t t h e c r o s s s e c t i o n o f a homogeneous c o n d u c t o r be s u b d i v i d e d i n t o a number o f f i l a m e n t s o f e q u a l c r o s s - s e c t i o n a 1 a r e a , as shown i n F i g u r e 1.1 . L e t t h e t o t a l number o f f i l a m e n t s be " g " . T h e n t h e d . c . r e s i s t a n c e o f t h e c o n d u c t o r i s g i v e n by t h e r e s i s t a n c e of one f i l a m e n t "R^" d i v i d e d by the t o t a l number of f i l a m e n t s , o r R d c - V * The i n s t a n t a n e o u s f i l a m e n t c u r r e n t " i " may be thought of as the sum o f t h e average f i l a m e n t c u r r e n t " i ^ " and the d e v i a t i o n A i from t h i s a v e r a g e , so t h a t i = i f + A i - 6 -1 1 A General Filament/ General f i l a m e n t F i g . 1 . 1 C o n d u c t o r s u b d i v i s i o n i n t o g f i l a m e n t s o f e q u a l c r o s s s e c t i o n a l a r e a a n d f i l a m e n t r e s i s t a n c e R ^ . B y a . c . r e s i s t a n c e i s m e a n t t h e r e s i s t a n c e o b t a i n e d b y d i v i d i n g t h e t o t a l o h m i c l o s s e s b y t h e s q u a r e o f t h e t o t a l c u r r e n t . R = Z [ R * ( i . + A i ) 2 ] a c g L f v f ' 1 ( 8 i f ) 2 R ^ - I ( i _ f + A i ) 2 w i t h t h e t e r m R d c 8 ^ 2 R = Z 1 + 2 l A i + 1 Z A i 2 a c 8 _ g R d c 8 8 i f 8 i f 2 R = 1 + 1 Z A i 2 a c R d c 8 if2 g Z A i = 0 ( b y d e f i n i t i o n , t h e s u m o f 8 t h e c u r r e n t d e v i a t i o n f r o m t h e a v e r a g e m u s t b e z e r o ) 2 1 Z A i b e i n g a l w a y s p o s i t i v e - 7 -T h e r e f o r e , u n l e s s A i i s z e r o i n every f i l a m e n t ( u n i f o r m c u r r e n t d i s t r i b u t i o n ) , the ac-dc r e s i s t a n c e r a t i o w i l l be g r e a t e r than one. T h i s proves t h a t : Rac > Rdc o r , what i s e q u i v a l e n t , t h a t t h e J o u l e l o s s e s a r e minimum f o r u n i f o r m c u r r e n t d i s t r i b u t i o n [ 1 ] . 1.2.2 P r o x i m i t y E f f e c t The a.c. c u r r e n t d i s t r i b u t i o n o v er the c r o s s s e c t i o n of a c o n d u c t o r depends not o n l y on the magnetic s e l f - f i e l d but a l s o on t h e f i e l d s g e n e r a t e d by c u r r e n t i n a d j a c e n t c o n d u c t o r s . The a d d i t i o n a l f i e l d component of n e i g h b o u r i n g c u r r e n t s i n c r e a s e s the a.c. r e s i s t a n c e i n most c a s e s . The change i n a.c. r e s i s t a n c e due t o c u r r e n t s i n o t h e r c o n d u c t o r s i s c a l l e d " p r o x i m i t y e f f e c t " . T h i s d i s t o r t i o n , ^ u n l i k e t h a t due t o s k i n e f f e c t , i s n o t s y m m e t r i c a l around the a x i s of symmetry of the c o n d u c t o r s ( i f the c o n d u c t o r i s c i r c u l a r ) . I n a t w o - w i r e l i n e , f o r i n s t a n c e , more c u r r e n t t e n d s t o f l o w e i t h e r on t h e s i d e where t h e c o n d u c t o r s f a c e each o t h e r or on the o p p o s i t e s i d e s . 1.3 A B r i e f E x p l a n a t i o n of the Method of Subconductors T h e a p p r o a c h u s e d i n t h i s t h e s i s s t a r t s w i t h t h e s u b d i v i s i o n o f a l l t h e c o n d u c t o r s i n t o " s u b c o n d u c t o r s " . By f i n d i n g the s e l f and mutual impedance of t h e s e s u b c o n d u c t o r s , and by b u n d l i n g s u b c o n d u c t o r s w i t h e q u a l v o l t a g e s b a c k i n t o t h e o r i g i n a l c o n d u c t o r s , t h e i mpedances o f t h e o r i g i n a l c o n d u c t o r s are o b t a i n e d . - 8 -S e v e r a l m e t h o d s h a v e b e e n b a s e d o n t h i s a p p r o a c h . T h e c o n c e p t s a r e b a s i c a l l y t h e s a m e , w i t h t h e m a i n d i f f e r e n c e b e i n g i n t h e c r o s s - s e c t i o n a l s h a p e o f t h e s u b c o n d u c t o r s . C o m e l l i n i e t a l d i v i d e t h e c o n d u c t o r s i n t o c y l i n d r i c a l s u b c o n d u c t o r s , G r a n e a u [ 1 ] d i v i d e s t h e m i n t o s q u a r e s , a n d L u c a s a n d T a l u k d a r i n t r o d u c e " e l e m e n t a l s " , w h i c h c o n s i s t o f f i l a m e n t s o f l a r g e r c r o s s s e c t i o n a l a r e a . I n t h i s l a s t w o r k t h e s h a p e m o s t c o m m o n l y u s e d i s o b t a i n e d b y d i v i d i n g t h e c o n d u c t o r f i r s t i n t o c i r c u l a r r i n g s a n d t h e n f u r t h e r i n a r a d i a l f o r m i n s u c h a w a y t h a t t h e e d g e s o f m o s t s u b c o n d u c t o r s a r e f o r m e d b y s t r a i g h t l i n e s a n d c i r c u l a r a r c s . T h e b a s i c i d e a i n t h i s a p p r o a c h i s t o m i n i m i z e t h e n u m b e r o f f i l a m e n t s . T h e c o m p u t a t i o n o f m u t u a l a n d s e l f i n d u c t a n c e s o f t h e " e l e m e n t a l s " i n t h e w o r k o f T a l u k d a r i s c o n s i d e r a b l y m o r e d i f f i c u l t t h a n i n t h e c a s e o f s i m p l e r f i l a m e n t s o f e x c l u s i v e l y s q u a r e o r c i r c u l a r c r o s s s e c t i o n . T h i s i s t h e r e a s o n w h y t h e s h a p e s o f t h e c r o s s s e c t i o n a l a r e a s o f t h e s u b c o n d u c t o r s , a r e l i m i t e d t o t h o s e f o r w h i c h i n d u c t a n c e s a r e r e a s o n a b l y e a s y t o o b t a i n . I n t h i s t h e s i s , t h e t h r e e m e t h o d s o f s u b d i v i s i o n a r e s t u d i e d a n d a c o m p a r i s o n b e t w e e n t h e m i s m a d e i n t e r m s o f a c c u r a c y , c o m p u t i n g t i m e a n d t h e a m o u n t o f s t o r a g e r e q u i r e m e n t s . T h e p e r f o r m a n c e o f e a c h m e t h o d i s c r i t i c a l l y a f f e c t e d b y t w o f a c t o r s : a ) T h e s i z e , s h a p e a n d d i s p o s i t i o n o f t h e f i l a m e n t s , t h a t i s t h e s c h e m e u s e d t o d i v i d e e a c h c o n d u c t o r i n t o f i l a m e n t s . - 9 -b)The a c c u r a c y of t h e f o r m u l a e employed t o d e t e r m i n e the " l o o p " i n d u c t a n c e s . The r e s u l t s of a l l methods a r e compared w i t h a t e s t case f o r w h i c h an a n a l y t i c a l s o l u t i o n i s known, such as f o r a c o a x i a l c a b l e w i t h o u t e a r t h r e t u r n . A n a l y t i c a l l y d e r i v e d g r o u n d r e t u r n f o r m u l a e a r e i n c o r p o r a t e d i n t o t h e model so t h a t no g r o u n d s u b d i v i s i o n i s n e c e s s a r y . T h i s r e d u c e s t h e number of sub c o n d u c t o r s , s t o r a g e r e q u i r e m e n t s and computing t i m e . O n c e t h e b e s t o f t h e m e t h o d s o u t l i n e d a b o v e was d e t e r m i n e d , i t was a p p l i e d t o t h e c a s e o f a p i p e type c a b l e . The e f f e c t o f s a t u r a t i o n i n t h e c a s e o f n o n - l i n e a r m a g n e t i c p i p e m a t e r i a l s was c o n s i d e r e d as w e l l S k i n and p r o x i m i t y e f f e c t s were a l s o s t u d i e d a t t h i s p o i n t , by o b t a i n i n g t h e c u r r e n t d i s t r i b u t i o n i n t h e p i p e and i n the i n t e r n a l c o n d u c t o r s . The c a s e o f s t r a n d e d c o n d u c t o r s was a l s o s t u d i e d w i t h t h i s t e c h n i q u e , w i t h o u t c o n s i d e r i n g t h e e f f e c t o f s p i r a l l i n g and the r e s u l t s were compared w i t h f i e l d t e s t d a t a and w i t h f o r m u l a e based on e x p e r i m e n t a l f i e l d p l o t t i n g . The i m p o r t a n c e o f t h i s w o rk l i e s n o t so much i n t h e d e v e l o p m e n t o f t h e t e c h n i q u e a s s u c h , f o r w h i c h t h e b a s i c c o n c e p t s a r e q u i t e s i m p l e , but i n t h e development of a computer program w h i c h can be used t o s o l v e some i m p o r t a n t p r o b l e m s f o r w h i c h o n l y e x p e r i m e n t a l s o l u t i o n s have been a v a i l a b l e so f a r . -10-2.THEORY AND FORMATION OF THE IMPEDANCE MATRIX 2.1 G e n e r a l Assumptions I n t h e model d e s c r i b e d b e l o w , each c o r e o f t h e c a b l e i s c o n s i d e r e d a s a m a i n c o n d u c t o r , a s a r e t h e s h e a t h , n e u t r a l c o n d u c t o r s a n d t h e a r m o u r . T h e s e m a i n c o n d u c t o r s a r e s u b d i v i d e d i n t o f i l a m e n t s . Three d i f f e r e n t c r o s s s e c t i o n a l a r e a s are c o n s i d e r e d f o r t h e s e f i l a m e n t s : - C i r c u l a r -Square - E l e m e n t a l As an e x a m p l e f o r t h e s u b d i v i s i o n i n t o f i l a m e n t s , F i g u r e 2.1 shows how a c o a x i a l c a b l e i s s u b d i v i d e d f o r e a c h c a s e . (a) C i r c u l a r (b) Square ( c ) E l e m e n t a l F i g . 2.1 S u b d i v i s i o n of the main c o n d u c t o r w i t h the t h r e e t y p e s of f i l a m e n t s . In t h i s p r o c e d u r e i t i s assumed t h a t : a ) T h e t r a n s m i s s i o n l i n e c r o s s s e c t i o n i s u n i f o r m , - 1 1 -t h a t i s , t h e e l e c t r o m a g n e t i c e f f e c t o f i r r e g u l a r i t i e s s u c h a s s p a c e r s i n c a b l e s a n d t o w e r s i n o v e r h e a d l i n e s i s n e g l i g i b l e . T h i s m e a n s t h a t e a c h s u b c o n d u c t o r i s u n i f o r m a n d h o m o g e n e o u s t h r o u g h o u t i t s l e n g t h . b ) C u r r e n t f l o w s l o n g i t u d i n a l l y i n t h e f i l a m e n t s . c ) T h e c u r r e n t d e n s i t y i s c o n s t a n t o v e r e a c h f i l a m e n t ' s c r o s s s e c t i o n . d ) T h e m a g n e t i c p e r m e a b i l i t y o f a s u b c o n d u c t o r i s c o n s t a n t t h r o u g h o u t t h e w h o l e c y c l e o f a l t e r n a t i n g c u r r e n t , b u t m a y b e d i f f e r e n t f r o m t h a t o f a n y o t h e r s u b c o n d u c t o r . e ) T h e c o n d u c t i v i t y o f e a c h f i l a m e n t i s c o n s t a n t ( t h o u g h n o t n e c e s s a r i l y t h e s a m e a s f o r a n y o t h e r f i l a m e n t ) . f ) A l l s u b c o n d u c t o r s a r e p a r a l l e l . 2 . 2 E q u a t i o n s F i r s t t h e c o n d u c t o r i s s u b d i v i d e d i n t o f i l a m e n t s . W e t h e n p r o c e e d t o d e t e r m i n e t h e r e s i s t a n c e a n d s e l f i n d u c t a n c e o f e a c h f i l a m e n t , a s w e l l a s t h e m u t u a l i n d u c t a n c e b e t w e e n f i l a m e n t s . T h e v a l u e o f t h e r e s i s t a n c e " r " o f e a c h f i l a m e n t i s e a s y t o o b t a i n b e c a u s e w e h a v e a s s u m e d t h a t t h e c u r r e n t d e n s i t y i s c o n s t a n t o v e r e a c h f i l a m e n t c r o s s s e c t i o n . T h e r e f o r e , i t i s o n l y a f u n c t i o n o f t h e r e s i s t i v i t y , l e n g t h a n d c r o s s s e c t i o n a l a r e a . - 1 2 -T h e I n d u c t a n c e d e t e r m i n a t i o n h o w e v e r , i s m o r e i n v o l v e d a n d r e q u i r e s t h a t a r e t u r n p a t h b e d e s i g n a t e d , a s u s e f u l v a l u e s o f i n d u c t a n c e s c a n o n l y b e d e f i n e d f o r c l o s e d c u r r e n t p a t h s . T h i s r e t u r n f i l a m e n t c a n b e o n e f i l a m e n t o f t h e c o n d u c t o r , o r a f i c t i t i o u s " r e t u r n p a t h " w h i c h a l l o w s m o r e f l e x i b i l i t y i n t h e m o d e l . T h e r e f o r e a f i c t i t i o u s r e t u r n p a t h i s c h o s e n h e r e . 2 . 2 . 1 C i r c u l a r F i l a m e n t s T o d e r i v e t h e l o o p i m p e d a n c e o f c i r c u l a r s u b c o n d u c t o r s , f i r s t c o n s i d e r t h e t w o l o o p s f o r m e d b y a n y t w o s u b c o n d u c t o r s " i " , " j " a n d t h e r e t u r n p a t h " q " a s s h o w n i n F i g u r e 2 . 2 F i g . 2 . 2 L o o p s f o r m e d b y t w o s u b c o n d u c t o r s w i t h a c o m m o n r e t u r n . W r i t i n g t h e l o o p e q u a t i o n f o r s u b c o n d u c t o r " i " g i v e s N Z M d i d t ( 2 . 1 ) -13-where - A v i i s t h e v o l t a g e d r o p p e r u n i t l e n g t h o f l i n e , R^,R^ are the d.c. r e s i s t a n c e s per u n i t l e n g t h o f c o n d u c t o r s i a n d . t h e r e t u r n p a t h q, r e s p e c t i v e l y , - i ^ , i j a r e the c u r r e n t s t h rough the s u b c o n d u c t o r s , - are the mutual i n d u c t a n c e s between l o o p s formed by s u b c o n d u c t o r s " 1 ' ^ " ^ " a n ( i r e t u r n path q, and - N i s the number of the s u b c o n d u c t o r s . F o r s t e a d y - s t a t e c o n d i t i o n s , t h e i n s t a n t a n e o u s v o l t a g e v and c u r r e n t i a r e r e p l a c e d by t h e i r p h a s o r v a l u e s , V and I , and d i / d t i s r e p l a c e d by j t o l . I n t h i s way, e a c h f i l a m e n t c a n be r e p r e s e n t e d by a c o n s t a n t lumped r e s i s t a n c e and i n d u c t a n c e , as shown i n F i g u r e 2.3, w i t h c o u p l i n g among t h e i n d u c t a n c e , b u t w i t h o u t c o u p l i n g among the r e s i s t a n c e s . conductor conductor < l a t n r n Path F i g . 2.3 A l u m p e d c o n s t a n t p a r a m e t e r m o d e l o f an N co n d u c t o r t r a n s m i s s i o n l i n e . To o b t a i n t h e s e l f a n d m u t u a l i n d u c t a n c e s b e t w e e n c o n d u c t o r s , t h e g e o m e t r y r e p r e s e n t i n g t h e c r o s s s e c t i o n s of c o n d u c t o r s " i " , " j " and the r e t u r n p a t h as w e l l as t h e d i s t a n c e s -14-between them i s needed, as shown i n F i g u r e 2.4, F i g . 2.4 Geometry of s u b c o n d u c t o r s . L e t us assume t h a t c u r r e n t i s o n l y i n j e c t e d i n t o s ubconductor " i " and the r e t u r n i s t h r o u g h "q". The m a g n e t i c f i e l d B a t a d i s t a n c e r f r o m any o f t h e su b c o n d u c t o r s i s g i v e n by Ampere's Law B = ^o 1 2 IT r (2.2) where I i s t h e c u r r e n t i n j e c t e d . Then t h e t o t a l f l u x p e r u n i t l e n g t h i n t h e e l e m e n t a l c y l i n d e r o f t h i c k n e s s dr i s g i v e n by 6\\> = B dr - u I dr o 2 IT r (2.3) The f l u x l i n k a g e of l o o p " j q " due t o t h e c u r r e n t i n " i " i s g i v e n by: r=D. i q u I dr = u I I n D. o o i q 2 TT r 2 TT D. r=D (2.4) -15-T h e r e i s a l s o a component o f t h e f l u x l i n k a g e w h i c h i s p r o d u c e d by t h e c u r r e n t t h r o u g h t h e r e t u r n p a t h " q " w h i c h i s g i v e n by: r=D . J q u I dr - u I In D. o o .jr r=r eq-q 27T r 2 77 r eq-q (2.5) where r = r e 0»25P ^ g u g e ( j t o a c c o u n t f o r the eq-q q i n t e r n a l f l u x i n the r e t u r n p a t h , w i t h b e i n g the r e l a t i v e p e r m e a b i l i t y o f the r e t u r n p a t h . The two f l u x e s a r e a d d i t i v e , so t h a t the t o t a l f l u x i s g i v e n by: . „ = a I In D . D. I j-T o _ir j r 2 IT r D. . eq-q i j The mutual i n d u c t a n c e between l o o p s " i " and " j " i s t h e n : (2.6) M. . = I . . „ = u„ I n D. D. i j yi.1-T M o i r i r I 2 TT r D. . eq-q i j ( 2 . 7 ) M i j " Hp ( I n D i r D i r + 0.25 M q ) 2 IT r D, . ^ A J (2.8) To d e r i v e t h e s e l f i n d u c t a n c e o f l o o p " i q " we c o n s i d e r t h e same c u r r e n t p a t h . The t o t a l f l u x l i n k a g e i s g i v e n by t h e f l u x l i n k a g e o f l o o p " i q " due t o t h e c u r r e n t I i n f i l a m e n t " i " p l u s t h e f l u x l i n k a g e of l o o p " i q " due t o t h e c u r r e n t I r e t u r n i n g t h r o u g h the r e t u r n p a t h " q " . The t o t a l f l u x l i n k a g e o f l o o p " i q " i s t hen g i v e n by: -16-4> i i - T D. J i q y ^ dr + T ^ " ± 2TT r r p i dr e q " q 2TT r (2.9) * i i - T y I In D 2 TT i3_ e q - i eq-q (2.10) L i i = In D, A3_ r . r e q - i eq-q (2.11) -0.25 u. r = r . e I e q - i l -0.25 y r = r e q eq-q q The f o r m u l a f o r t h e s e l f i n d u c t a n c e t h e r e f o r e becomes: (2.12) M.. = L., = y ( In D. +0.25 y.+0.25 y ) l i l i 9 / i q l q 2 TT- r r . q i (2.13) W r i t i n g t h e l o o p e q u a t i o n s f o r a l l s u b c o n d u c t o r s w i t h r e t u r n t h r o u g h r e t u r n p a t h q l e a d s t o t h e f o l l o w i n g s y s t e m o f l i n e a r e q u a t i o n s : " w l l " • • •AVNm. Z l l l l ' ' ' Z l l l n 1 - * 1 Z 1 1 N 1 * • Z l l N m Z l n l l ' ' , Z l n l n ' ^ l n N l * ' ZlnNm ZN111* ' * Z N l l n i : : i : : : - Z N m l l * " ' ZNmln i . . i . . . NmNm 11 In N l ^Nm-1 (2.14) -17-[ A V] = [ Z ] [ l ] (2.15) where: A V ^ = v o l t a g e drop on subconductor " 1 " of co n d u c t o r "k", * k l = c u r r e n t through s u b c o n d u c t o r " 1 " of c o n d u c t o r "k", ^ k l m i = m u t u a ^ - impedance between the l o o p s formed by s u b c o n d u c t o r s " 1 " and " i " o f main c o n d u c t o r s "k" and "n" r e s p e c t i v e l y . The r e s i s t a n c e s and i n d u c t a n c e s i n t h e [ Z ] m a t r i x a r e c o n s t a n t , but s i n c e t h e c u r r e n t d i v i s i o n among s u b c o n d u c t o r s changes w i t h f r e q u e n c y , s k i n and p r o x i m i t y e f f e c t s a r e a c c o u n t e d f o r . The l o c a t i o n and shape of t h e " R e t u r n P a t h " a f f e c t s t h e v a l u e s of t h e l o o p i n d u c t a n c e s . However, t h e s e e f f e c t s c a n c e l out when t h e z e r o c u r r e n t c o n s t r a i n t i s imposed on t h e " R e t u r n P a t h " , as e x p l a i n e d l a t e r . The b u n d l i n g o f f i l a m e n t s i n s i d e each c o n d u c t o r w i l l be e x p l a i n e d l a t e r as w e l l , b e c a u s e i t i s a p r o c e d u r e w h i c h i s common t o a l l s u b d i v i s i o n methods. 2.2.2 Square F i l a m e n t s The c o n d u c t o r s i n v o l v e d i n t h e t r a n s m i s s i o n s y s t e m a r e now d i v i d e d i n t o s q u a r e s , as shown i n F i g u r e 2.1. B e c a u s e o f t h i s new t y p e o f g e o m e t r y we c a n n o t a p p l y t h e same f o r m u l a s o b t a i n e d f o r t h e c a s e o f c i r c u l a r f i l a m e n t s , s i n c e t h e f i e l d s p r o d u c e d by c u r r e n t s i n s q u a r e f i l a m e n t s a r e d i f f e r e n t f r o m t h o s e c r e a t e d by c i r c u l a r c r o s s s e c t i o n f i l a m e n t s . -18-Th e c l a s s i c a l c o n c e p t o f m u t u a l i n d u c t a n c e between two c l o s e d c u r v e s was f i r s t s u g g e s t e d by F.E. Neumann i n 1845 b e f o r e t h e p u b l i c a t i o n o f M a x w e l l ' s f i e l d t h e o r y . I t i s e m b o d i e d i n Neumann's (see r e f e r e n c e [ 1 ] ) well-known law M = y 4 Tf cos dm dr . r (2.16) m n where the symbols m,n s t a n d f o r two c l o s e d c u r v e s w i t h dm b e i n g a s h o r t e l e m e n t o f m and dn a s h o r t e l e m e n t o f n. The d i s t a n c e b e t w e e n t h e s e two e l e m e n t s i s d e n o t e d by r , a n d i s t h e a n g l e of i n c l i n a t i o n between dm and dn. I f t h i s f o r m u l a i s a p p l i e d t o t w o r e c t a n g u l a r arrangements ( F i g u r e 2.5) i n which the l o o p s are p e r p e n d i c u l a r to each o t h e r and t h e segments CD and HG are l o c a t e d i n f i n i t e l y f a r away, i t i s p o s s i b l e t o o b s e r v e t h a t f o r the l i n e elements dm and dn l o c a t e d on the r e c t a n g u l a r f i l a m e n t s AB and EF, the i n t e g r a n d has f i n i t e magnitude. However the i n t e g r a n d v a n i s h e s f o r l i n e element c o m b i n a t i o n s of any o t h e r p a i r of s i d e s . T h i s proves t h a t i n t h i s case t h e mutual i n d u c t a n c e s a c t i n g between p a r a l l e l s t r a i g h t l i n e s may be c a l c u l a t e d by Neumann's f o r m u l a w i t h o u t r e g a r d t o the r e t u r n c i r c u i t . -19-Loop n F i g . 2.5 R e c t a n g u l a r l o o p s - Neumann Formulas Then t h e i n t e g r a l o v e r t h e c l o s e d l o o p s r e d u c e s t o t h e l i n e i n t e g r a l over AB and EF M . I1 l% y cos dm dn (2.17) o 0 4 7T r T h i s i n t e g r a l was s o l v e d by Sommerfeld and t h e s o l u t i o n i s g i v e n by M = y ( % In % + A 2 + d 2 - A 2 +d 2 + d) m,n N  2 ir d (2.18) When i >> d (d i s the d i s t a n c e between AB and EF) the f o r m u l a s i m p l i f i e s t o the a p p r o x i m a t i o n M * y In 2 I _JLtiL H/m I 2 TT e d (2.19) For vacuum and a i r the p e r m e a b i l i t y i s V = y Q = 4 T f l O - ^ Then M = 2 1 0 " 7 In 2 A H/m m, n  A e d (2.20) -20-I n t h i s c a s e i t i s not n e c e s s a r y t o c r e a t e a f i c t i t i o u s r e t u r n p a t h . F o r t h e c a l c u l a t i o n o f t h e s e l f i n d u c t a n c e , t h e "mean g e o m e t r i c d i s t a n c e c o n c e p t " i s a p p l i e d . T h i s c o n c e p t i s based on t h e f a c t t h a t t h e s e l f - i n d u c t a n c e of a homogeneous l i n e a r c o n d u c t o r , c a r r y i n g u n i f o r m l y d i s t r i b u t e d c u r r e n t , i s e q u a l t o the m u t u a l i n d u c t a n c e o f two t h i n w i r e s , s e p a r a t e d by the mean g e o m e t r i c d i s t a n c e of the c o n d u c t o r c r o s s s e c t i o n . L. . i s t h e n g i v e n by e q u a t i o n ( 2 . 2 1 ) w h e r e d i s t h e GMD i f 1 ( g e o m e t r i c mean d i s t a n c e ) of a square d= 0.44705 a w i t h " a " b e i n g t h e l e n g t h o f one s i d e o f t h e s q u a r e . Then L.- . = 2 1 0 ~ 7 In 2 1 H/m l 0.44705a e (2.21) Once t h e i n d u c t a n c e s a r e d e t e r m i n e d , i t i s p o s s i b l e to w r i t e t h e same s y s t e m o f l i n e a r e q u a t i o n s as f o r t h e c a s e o f c i r c u l a r f i l a m e n t s , [ AV] = [ Z ] [ I ] 2.2.3 E l e m e n t a l s I n a l l d i s c r e t e v a r i a b l e p r o c e d u r e s , t h e q u a n t i z a t i o n e r r o r depends on t h e i n t e r v a l s i z e and on t h e r a t e o f change of th e dependent v a r i a b l e s w i t h r e s p e c t t o the in d e p e n d e n t ones. The h i g h e r t h i s r a t e , t h e s m a l l e r an i n t e r v a l i s r e q u i r e d . T h i s i s i m p o r t a n t t o k e e p i n m i n d w i t h t h i s t y p e of f i l a m e n t , b e c a u s e t h e y u s u a l l y have a l a r g e r c r o s s s e c t i o n a r e a t h a n s q u a r e s o r c i r c u l a r f i l a m e n t s . The a s s u m p t i o n o f u n i f o r m c u r r e n t d i s t r i b u t i o n a c r o s s t h e e l e m e n t a l can t h e r e f o r e l e a d to e r r o r s i n zones where the r a t e of change of c u r r e n t i s ve r y h i g h . -21-F o r t h e example of a c o a x i a l c a b l e t h e c u r r e n t d e n s i t y and i t s r a t e o f change v a r y e x p o n e n t i a l l y w i t h d i s t a n c e a l o n g r a d i i and a r e c o n s t a n t on c i r c l e s t h a t a r e c o n c e n t r i c w i t h t h e c o n d u c t o r s . I n t h i s c a s e an optimum arrangement ( i n terms of the numbers of f i l a m e n t s needed t o produce a g i v e n l e v e l o f a c c u r a c y ) w o u l d be c y l i n d r i c a l f i l a m e n t s whose t h i c k n e s s e s d e c r e a s e e x p o n e n t i a l l y to t h e edge o f t h e i n n e r c o n d u c t o r and i n c r e a s e s f r o m t h e i n n e r edge of t h e o u t e r c o n d u c t o r t o i t s o u t e r edge. F o r a s y m m e t r i c a l c o n d u c t o r a r r a n g e m e n t s where i t i s not p o s s i b l e t o p r e d i c t e x a c t l y where th e h i g h e s t r a t e s o f change of c u r r e n t d e n s i t y w i l l o c c u r , o t h e r t y p e s o f e l e m e n t a l c r o s s s e c t i o n a l s h a p e s a r e n e e d e d . U s u a l l y t h e e l e m e n t a l s h a p e s s h o w n i n F i g u r e 2.6 a r e e n o u g h t o f o r m a n y o p t i m a l arrangement. T h e c o m p u t a t i o n o f e l e m e n t a l i n d u c t a n c e s i s v e r y d i f f i c u l t , s i n c e t h e r e a r e no a n a l y t i c a l f o r m u l a e a v a i l a b l e . F i g . 2.6 " E l e m e n t a l " c r o s s - s e c t i o n a l shapes. L u c a s and T a l u k d a r d e v e l o p e d some a p p r o x i m a t i o n s f o r t h e s e i n d u c t a n c e c a l c u l a t i o n s , w h i c h are based on the a s s u m p t i o n -22-t h a t t h e c o n d u c t o r s and i n t e r v e n i n g s p a c e s h a v e t h e same c o n s t a n t p e r m e a b i l i t y . T h e i r a pproach i s e x p l a i n e d below, and i s d i v i d e d i n t o t h r e e p o i n t s : a) E l e m e n t a l I n d u c t a n c e s GMD ( G e o m e t r i c Mean D i s t a n c e ) : C o n s i d e r two f i l a m e n t s AB and CD as i n F i g u r e 2.7. l it ds F i g . 2.7 F i l a m e n t s - " E l e m e n t a l " Impedance C a l c u l a t i o n . The f l u x l i n k a g e of f i l a m e n t CD due t o the c u r r e n t i n AB i s g i v e n by Neumann's law from e q u a t i o n [ 2 . 2 0 ] , T|> - 2 1 0 " 7 H dAj [ l n ( 2 A ) -1] d (2.22) 3 = c u r r e n t d e n s i t y C o n s i d e r a p a i r of p a r a l l e l m o n o l i t h i c f i l a m e n t s , w i t h a r b i t r a r y c r o s s s e c t i o n a l a r e a s A^ and A2 as shown i n F i g u r e 2 . 8 . I n o r d e r t o o b t a i n t h e f l u x l i n k a g e o f A2 d u e t o A j , t h e s u p e r p o s i t i o n t h e o r e m i s u s e d t o d e t e r m i n e t h e f l u x l i n k a g e ipAj, dA2 produced by the e n t i r e c u r r e n t i n t h e m o n o l i t h i c s u b c o n d u c t o r A^. W i t h t h e a s s u m p t i o n of c o n s t a n t p e r m e a b i l i t y t h e f l u x l i n k a g e i s g i v e n by: See f i g u r e 2.8 -23-i|>AlfdA2 = 2 10 - 7 I P( i ) A l [ In ( 2 I ) - l ] dA 1 (2.23) Obv i o u s l y the f l u x l i n k a g e of dA 2 v a r i e s w i t h i t s l o c a t i o n w i t h i n A 2« T h e r e f o r e a v a r y i n g c u r r e n t i n w i l l i n d u c e d i f f e r e n t v o l t a g e s i n d i f f e r e n t p a r t s of But s i n c e t h e i d e a i s t o s i m u l a t e e a c h f i l a m e n t a s one l u m p e d r e s i s t a n c e and i n d u c t a n c e , i t becomes n e c e s s a r y t o a p p r o x i m a t e each q u a n t i t y t h a t v a r i e s over the c r o s s s e c t i o n of A 2 by a s i n g l e r e p r e s e n t a t i v e v a l u e . I n t h i s case the average i s used. Then: - 7 M 1 , A 2 = 2 10 ' I ( i ) 1 A l A 2 f [ In ( 2 i ) -1 ]dA dA An A, =2 1 0 " 7 I i ([ In 22 I) -1] - In G) where In G = A l A 2 ' In d dAj dA 2 A 2 A 1 * A l t A 2 = 2 10 - 7 i [ In ( 2 H ) ] G e (2.24) G i s the GMD (Geom e t r i c Mean D i s t a n c e ) -24-F i g . 2.8 P a r a l l e l m o n o l i t h i c f i l a m e n t s b) Loop I n d u c t a n c e I n t h i s c a s e a f i c t i t i o u s r e t u r n p a t h i s u s e d t o c a l c u l a t e t h e l o o p i n d u c t a n c e . A p p l y i n g f o r m u l a ( 2.24) t o t h r e e p a r a l l e l f i l a m e n t s i , j , r and f o r m i n g two l o o p s " i r " , " j r " , t h e f l u x l i n k a g e o f l o o p " j r " due t o t h e c u r r e n t t h r o u g h l o o p " i r " c a n be c a l c u l a t e d by a p p l y i n g t h e s u p e r p o s i t i o n t h e o r e m , \\). = ( ip. . - i|>. j r v T i j y i r r i r r ' (2.25) = 2 10 ~ 7 i I [ In (2 I ) - I n ( 2 £)-ln(2 A) + l n ( 2 A ) ] - 2 10 - 7 i r r i r r M I i ( In ( G. G. ) ) _ _ x r _ _ i r _ / 7 G G. . r r l j - = 2 1 0 " 7 ( l n ( G. G. )) j r v v i r i r y / I i G G. . r r l j (2.26) c) N u m e r i c a l Computation of GMD The GMD f o r f i l a m e n t s ( b ) , ( c ) o f F i g u r e 2.6 a r e w e l l known and t h e i r f o r m u l a s a r e easy t o f i n d i n hand books . T h e r e f o r e , o n l y the c a l c u l a t i o n of t h e GMD of f i l a m e n t s of the type (a) -25-w i l l be d i s c u s s e d i n more d e t a i l h e r e . S e v e r a l d i f f e r e n t s i t u a t i o n s must be c o n s i d e r e d : a) E l e m e n t a l s w h i c h a r e r e l a t i v e l y f a r a p a r t as shown i n F i g u r e 2.9: I n t h i s c a s e , e a c h l o n g n a r r o w e l e m e n t a l i s d i v i d e d i n t o f o u r s e c t o r s . P a i r s of p o i n t s a r e s y m m e t r i c a l l y p l a c e d about the l i n e s between s e c t o r s a t an a n g u l a r d i s t a n c e o f f A/6 where A i s t h e a n g l e spanned by t h e e l e m e n t a l , and f=0.564 i s a f a c t o r c h o s e n by L u c a s and T a l u k d a r t o m i n i m i z e t h e e r r o r . The GMD i s t a k e n t o be t h e g e o m e t r i c mean o f t h e 36 d i s t a n c e s between t h e s e t o f s i x p o i n t s i n one e l e m e n t a l and t h e s e t i n t h e o t h e r . T h i s a p p r o x i m a t i o n does not work w e l l f o r c a l c u l a t i n g t h e mutual i n d u c t a n c e between e l e m e n t a l s t h a t a r e v e r y c l o s e , such as t h o s e of F i g u r e 2.10 and f o r c a l c u l a t i n g the s e l f GMD. b) S e l f GMD: The s e l f GMD o f a r e c t a n g l e i s 0.2335(&+w). T h i s e x p r e s s i o n , w i t h a s l i g h t change i n t h e c o n s t a n t t o 0.2315, i s a good a p p r o x i m a t i o n f o r l o n g narrow c u r v e d e l e m e n t a l s . F i g . 2.9 " E l e m e n t a l s " r e l a t i v e l y f a r a p a r t . -26-F i g . 2.10 " E l e m e n t a l s " which a r e c l o s e t o g e t h e r . c ) E l e m e n t a l s which are very c l o s e t o g e t h e r : The mutual GMD between t h e p a i r o f s t r a i g h t l i n e s of F i g u r e 2.11 i s g i v e n by In (GMD) = [ ( X 2 2 + d 2 ) l n ( X 2 2 +d 2) + ( X X 2 + X 2 2 ) 2 n A 2 n Jl - ( X 1 2 + d 2 ) l n ( X 1 2 + d 2 ) + 2 d X 2 t a n _ 1 ( X 2 ) 2 n I n I d -(2 d X x ) t a n ~ 1 ( X 1 ) + ( d 2 ) l n ( d 2 + X : 2 ) -1] n I d n£ d 2+x 2 A (2.27) where X^ = & - n X 2 = A + n 2 2 (2.28) T h i s e x p r e s s i o n can be adapted t o t h e c a l c u l a t i o n o f the GMD b e t w e e n t h e e l e m e n t a l o f F i g u r e 2.11 by r e p l a c i n g t h e s t r a i g h t l i n e s l e n g t h , & and n w i t h t h e l e n g t h s o f the a r c s a l o n g t h e " c e n t e r s " o f t h e e l e m e n t a l s , and m u l t i p l y i n g by 0.955 t o compensate f o r c u r v a t u r e and n o n - z e r o t h i c k n e s s . I n t h i s t h e s i s i t was found t h a t b e t t e r r e s u l t s a r e o b t a i n e d u s i n g 0.833 i n s t e a d of 0.955. -27-F i g . 2.11 A p a i r of p a r a l l e l s t r a i g h t l i n e s . Once t h e i n d u c t a n c e s have been c a l c u l a t e d , t h e impedance m a t r i x i s formed. N o r m a l l y , t h e l a r g e i m p e d a n c e m a t r i x i s n o t o f d i r e c t i n t e r e s t . I n s t e a d , a m a t r i x r e l a t i n g v o l t a g e s t o c u r r e n t s i n the main c o n d u c t o r s i s needed . T h i s i s d i s c u s s e d n e x t . - 2 8 -3 .REDUCT ION OF THE LARGE IMPEDANCE MATRIX T w o c o n d i t i o n s a r e u s e d t o r e d u c e t h e [ Z ] m a t r i x : a ) T h e v o l t a g e d r o p s a l o n g s u b c o n d u c t o r s f o r m i n g t h e s a m e m a i n c o n d u c t o r a r e e q u a l V .,=v =v . J l J 2 j n ( 3 < 1 ) b ) T h e c u r r e n t i n a n y m a i n c o n d u c t o r i s t h e s u m o f t h e c u r r e n t s i n t h e s u b c o n d u c t o r s i n t o w h i c h i t i s d i v i d e d J J l J 2 j n ( 3 > 2 ) T h e r e f o r e , t h e s y s t e m o f e q u a t i o n s c a n b e w r i t t e n i n t h e f o l l o w i n g f o r m : Z l l 1 1 • * Z l l l n Z l n l l * " Z l n l n ' k i l l < - Z k m l l * ' Z k m k m Z l l k l ' * Z l l k m 1 1 I n ( 3 . 3 ) T h e m a t r i x r e d u c t i o n i s a c h i e v e d a s f o l l o w s : a ) S u b t r a c t t h e f i r s t e q u a t i o n o f e a c h c o n d u c t o r f r o m t h e s u b s e q u e n t e q u a t i o n s o f t h a t m a i n c o n d u c t o r . T h i s l e a v e s t h e l e f t - h a n d s i d e o f t h e o t h e r e q u a t i o n s e q u a l t o z e r o . b ) I f t h e t o t a l c u r r e n t t h r o u g h t h a t c o n d u c t o r i s s u b s t i t u t e d i n t o t h e f i r s t e q u a t i o n o f e a c h c o n d u c t o r ( i . e . , i n r o w 1 ) , a n e r r o r o f a d d i n g Z ^ J J + * * " * * Z 1 1 1 1 ^ l n i s m a < * e * n -29-t h i s c a s e . C o r r e s p o n d i n g e r r o r s a r e i n t r o d u c e d i n t o a l l the o t h e r e q u a t i o n s . T h e s e e r r o r s a r e r e m o v e d by s u b t r a c t i n g t h e f i r s t c o l u m n c o r r e s p o n d i n g to each main c o n d u c t o r from t h e subsequent columns of t h a t main c o n d u c t o r . T h e s e c h a n g e s p r o d u c e a s e t o f l i n e a r e q u a t i o n s e x p r e s s i n g t h e v o l t a g e s on t h e c o n d u c t o r s i n terms o f t h e t o t a l c u r r e n t i n t h e s e c o n d u c t o r s , 0 • _ 0 '1111 J k l l l 'kmll . 5 l l l n l l k l r _ 2 - Z kmqn kmqn k l q n '11km Z k l k l ' ? k l k m ? k m k l * ^ kmkm-: l 12 k l Z k m q l + Z k l q l f o r m,n £l (3.4) (3.5) f o r m =1 n ^ l ^ k l q n ~ Z k l q n Z q l q l f o r n =1 m^l C k m q l = Z k m q l " Z k l q l (3.7) The r e s t o f the elements remains unchanged. 3.1 R e d u c t i o n of the Large Impedance M a t r i x The e q u a t i o n s a r e r e a r r a n g e d f o r t h e r e d u c t i o n p r o c e s s , by i n t e r c h a n g i n g rows and co l u m n s , so t h a t t h e f i n a l m a t r i x has -30-the form 0 0 0 0 Z l 1 1 1 * , 5 l l k l * ' k i l l * * ? k l k m -I L^k -or i n a b b r e v i a t e d form 0 V A C B Z I I (3.8) (3.9) R e f e r e n c e [10] p r o v i d e s a v e r y e f f i c i e n t way f o r r e d u c i n g t h i s s y s t e m o f e q u a t i o n s , u s i n g G a u s s i a n e l i m i n a t i o n . By e l i m i n a t i n g t h e v a r i a b l e s up t o the d i a g o n a l i n t h e rows of s e t A, and up t o t h e l a s t c o l u m n o f C i n t h e rows o f s e t C, t h e m a t r i x t a k e s on the form nonzero " 0 ' 0 0 0 = ~7l - V 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 nonzero -k-1 (3.10) From t h i s , the reduced system of e q u a t i o n s becomes u 'k -» ^ k l Z l l * * Z l k Z, , . . Z kk -1 (3.11) -31-As p o i n t e d o ut b e f o r e , when a f i c t i t i o u s r e t u r n p a t h e x i s t s , as i n the case of c i r c u l a r and " e l e m e n t a l s " f i l a m e n t s , the l o c a t i o n and shape of the " r e t u r n p a t h " a f f e c t the v a l u e s of the l o o p i n d u c t a n c e . T h e r e f o r e the v a l u e s i n the re d u c e d [Z] m a t r i x w i l l depend on t h e s e two p a r a m e t e r s . However, t h e s e e f f e c t s c a n c e l out when the z e r o - c u r r e n t c o n s t r a i n t i s imposed on the r e t u r n path. T h e o r e t i c a l l y then, the r e t u r n path c o u l d be placed anywhere and have any shape. T h i s c o n s t r a i n t of z e r o - c u r r e n t i m p l i e s t h a t the sum of a l l the c u r r e n t s t h r o u g h the main c o n d u c t o r s i s z e r o , t h a t i s Z I. = 0 i-1 1 S u b s t i t u t i n g t h i s c o n d i t i o n i n t o e q u a t i o n (3.11) g i v e s Z l l * * , Z l k i _ z k l . . . z k k J (3.12) Z l l " Z i k ' ' ' , Z l k - l Z l k Z, , - Z , . .Z k k - l ~ Z k k J i-^k-l-J *-"kl "kk* * (3.13) I f c o n d u c t o r k r e p r e s e n t s the l o c a l ground ( o r n e u t r a l c o n d u c t o r or the sh e a t h ) w i t h r e s p e c t to which a l l v o l t a g e s are measured, then i t s e q u a t i o n can be e l i m i n a t e d by s u b t r a c t i n g the e q u a t i o n f o r from a l l the o t h e r e q u a t i o n s , thus p r o d u c i n g a reduced system of equations given by: -32-w i t h ~ V, -V, 1 k • • = L. k-1 k-'11 * -' l k - 1 • • •* • • • * Z k l Z k - l k - l J (3.14) Z. . = Z. . + Z, , - 2 Z, . i j i j k k k i (3.15) T h i s e q u a t i o n r e l a t e s v o l t a g e s f r o m phase to g r o u n d to the phase c u r r e n t s . - 3 3 -4. RETURN PATH IMPEDANCE T h e r e a r e m a i n l y t h r e e p o s s i b l e r e t u r n p a t h s f o r t h e c u r r e n t i n any t r a n s m i s s i o n system: - R e t u r n i n n e u t r a l c o n d u c t o r s o n l y - R e t u r n i n ground o n l y - R e t u r n i n ground and n e u t r a l c o n d u c t o r s F o r t h e r e t u r n t h r o u g h n e u t r a l c o n d u c t o r s , e a c h one of them ( s h e a t h , n e u t r a l c o n d u c t o r s or p i p e ) i s c o n s i d e r e d as a main c o n d u c t o r and s u b d i v i d e d i n t o f i l a m e n t s . Once t h e Z m a t r i x i s f o r m e d , t h e n e u t r a l c o n d u c t o r s o r s h e a t h s can be e l i m i n a t e d i n t h e r e d u c t i o n p r o c e s s , i f so d e s i r e d , t o o b t a i n t h e impedance m a t r i x w h i c h r e l a t e s t h e v o l t a g e s f r o m p h a s e t o n e u t r a l t o t h e phase c u r r e n t s . Any o t h e r c o n d u c t o r can a l s o be e l i m i n a t e d i n t h i s way p r o v i d e d t h a t t h e r e i s z e r o v o l t a g e on i t , o r t h a t i t i s c o n n e c t e d i n p a r a l l e l w i t h one or more o f t h e o t h e r n e u t r a l c o n d u c t o r s . For t h e ground r e t u r n t h e r e a r e s e v e r a l models w h i c h can be used: a)Model I t S u b d i v i d i n g the ground. The ground i s s u b d i v i d e d i n t o l a y e r s of s u b c o n d u c t o r s as s h o w n i n F i g u r e 4.1 , t o r e p r e s e n t t h e s o i l by a s e t o f e l e m e n t a r y c o n d u c t o r s whose d i a m e t e r i n c r e a s e s as t h e d i s t a n c e f r o m phase c o n d u c t o r s i n c r e a s e s . The r e a s o n f o r t h e change i n d i a m e t e r i s t h e f a c t t h a t t h e c u r r e n t d e n s i t y i n t h e s o i l d e c r e a s e s a s o n e m o v e s f a r t h e r away f r o m t h e c a b l e s . -34-b ) M o d e l I I ; U s i n g g r o u n d r e t u r n f o r m u l a e d i r e c t l y w i t h t h e s u b c o n d u c t o r s . A v e r y s i m p l e m o d e l w h i c h u s e s t h e a n a l y t i c a l g r o u n d r e t u r n f o r m u l a e ( C a r s o n , P o l l a c z e k . W e d e p o h l a n d W i l c o x , o r s i m i l a r f o r m u l a e ) t r e a t s e a c h s u b c o n d u c t o r a s an i n s u l a t e d c o n d u c t o r w i t h t h e r e t u r n l o o p t h r o u g h t h e g r o u n d and u s e s t h e a v a i l a b l e g r o u n d r e t u r n f o r m u l a e t o c a l c u l a t e t h e s e l f a n d m u t u a l i m p e d a n c e s . c ) M o d e l I I I ; R e p r e s e n t i n g t h e g r o u n d a s one u n d i v i d e d c o n d u c t o r I n t h i s m o d e l , t h e g r o u n d i s c o n s i d e r e d as one a d d i t i o n a l c o n d u c t o r ( n o t s u b d i v i d e d i n t h i s c a s e ) and t h e m u t u a l i m p e d a n c e b e t w e e n t h e g r o u n d and s u b c o n d u c t o r s a r e c a l c u l a t e d u s i n g g r o u n d r e t u r n f o r m u l a e . T h e m a i n a d v a n t a g e o f M o d e l I I I i s t h a t t h e m o r e c o m p l i c a t e d g r o u n d r e t u r n f o r m u l a e n e e d o n l y be u s e d i n one row and one column o f t h e [ Z ] m a t r i x , w h i l e w i t h M o d e l I I t h e g r o u n d r e t u r n f o r m u l a e a f f e c t a l l t h e t e r m s o f t h e i m p e d a n c e m a t r i x . W i t h M o d e l I t h e number o f f i l a m e n t s must be f a i r l y h i g h , w h i c h i n c r e a s e s t h e c o m p u t a t i o n t i m e . I n t h i s t h e s i s M o d e l I I I was s e l e c t e d f o r i t s s i m p l i c i t y , a l t h o u g h i t i s t h e l e a s t a c c u r a t e m o d e l due t o t h e a p p r o x i m a t i o n made. I t i s i m p o r t a n t t o remember, h o w e v e r , t h a t an a n a l y t i c a l c a l c u l a t i o n i n v o l v i n g g r o u n d r e t u r n w i l l n e v e r be e x a c t anyhow, b e c a u s e t h e g r o u n d r e s i s t i v i t y i s n e i t h e r w e l l known nor u n i f o r m -35-F i g . 4 . 1 S u b d i v i s i o n o f t h e s o i l 4 . 1 A n a l y t i c a l G r o u n d R e t u r n F o r m u l a e E q u a t i o n s f o r g r o u n d r e t u r n i m p e d a n c e s w e r e d e r i v e d b y C a r s o n f o r o v e r h e a d t r a n s m i s s i o n l i n e s . T h e y c a n b e a p p l i e d t o u n d e r g r o u n d s y s t e m s a s w e l l i f t h e i m a g e s o f t h e c o n d u c t o r s a r e p l a c e d a b o v e t h e g r o u n d s u r f a c e a t h e i g h t s e q u a l t o t h e d e p t h o f b u r i a l , i n s t e a d o f b e l o w t h e s u r f a c e a s i n t h e c a s e o f o v e r h e a d l i n e s . C a r s o n ' s f o r m u l a f o r m u t u a l i m p e d a n c e s i s : Z 1 2 = j 2 w l n ( jj^) + 4 en J ( / o 2 + j - o ) e " ( h l ' + h 2 ' ^ o s x ' a d a w h e r e 0 ( 4 . 1 ) P " = / ( h l + h 2 ) 2 + X2 p ' = / ( h l - h 2 ) Z + X2 h i ' = h l / ~ a h 2 ' = h 2 / ~ a x ' = x / a a = 4TT X w h l , h 2 d e p t h o f b u r i a l o f c o n d u c t o r s 1 a n d 2 r e s p e c t i v e l y . -36-O n l y t h e m u t u a l impedance f o r m u l a i s g i v e n h e r e , because C a r s o n ' s f o r m u l a s f o r o v e r h e a d t r a n s m i s s i o n l i n e s c a n n o t be a p p l i e d f o r t h e c a l c u l a t i o n o f t h e s e l f impedance o f underground c a b l e s . I n a l a t e r p a p e r C a r s o n showed t h a t f o r c o n d u c t o r s b u r i e d u n d e r g r o u n d , t h e v a r i a t i o n o f g r o u n d r e t u r n i m p e d a n c e w i t h d i s t a n c e b e l o w t h e e a r t h s u r f a c e i s r e l a t i v e l y s m a l l f o r t h e u s u a l d e p t h s o f b u r i a l and t h a t t h e ground r e t u r n impedance (Zg) can be c a l c u l a t e d as w i t h Z - ( 1 + C) z ° 8 8 (4.2) Z ° = g r o u n d r e t u r n i m p e d a n c e i f t h e e a r t h were t o e x t e n d i n d e f i n i t e l y i n a l l d i r e c t i o n s around t h e c o n d u c t o r so t h a t c i r c u l a r symmetry e x i s t s . C = c o r r e c t i o n f a c t o r w h i c h a c c o u n t s f o r t h e f a c t t h a t t h e c o n d u c t o r i s l o c a t e d n e a r t h e g r o u n d s u r f a c e . 1 (4.3) 0.5 K 0 ( a ' j v ^ j ) F u r t h e r m o r e , when a' = a/~~ct (a = e x t e r n a l r a d i u s ) i s ve r y s m a l l compared t o u n i t y we can approximate C by: C = 1 2 l n ( l / a ' ) -37-Z ° = m 1 K (mr) 8 o 2 r K ^ m r ) m =/ j u u)/ p K ,K, a r e m o d i f i e d B e s s e l f u n c t i o n s , w i t h o' 1 ' r= i n t e r n a l r a d i u s of t h e e a r t h ( i . e . o u t e r r a d i u s o f the c o n d u c t o r i n s u l a t i o n ) P= ground r e s i s t i v i t y w = 2 TT f , f = f r e q u e n c y u s magnetic p e r m e a b i l i t y of the ground. E q u a t i o n s f o r c a l c u l a t i n g t h e s e l f and m u t u a l impedance of underground c o n d u c t o r s have a l s o been d e r i v e d by P o l l a c z e k i n t h e f o r m of i n f i n i t e s e r i e s . C l o s e d - f o r m a p p r o x i m a t i o n s to the s e l f and m u t u a l impedance o f u n d e r g r o u n d c o n d u c t o r s , w h i c h a r e v a l i d f o r a wide range of v a l u e s of t h e p a r a m e t e r s i n v o l v e d , have been d e r i v e d by Wedepohl & W i l c o x . These e q u a t i o n s a r e : Z = j H w ( - l n ( Ymr ) +0.5-4 mh) /m 3 2 TT 2 3 (4.4) Z = j u y ( _ l n ( Ym D i k ) +0.5 - 2 m l ) fl /m 2 TT 2 3 (4.5) where Z g , Z ^ a r e s e l f and mutual impedance of the ground r e t u r n p a t h , r e s p e c t i v e l y , w i t h Y = E u l e r ' s c o n s t a n t = 0.5772157 h = depth of b u r i a l of c o n d u c t o r ( m e t r e s ) 1 = sum of depth of b u r i a l of c o n d u c t o r s i and k (metres) ^ i k = d i s t a n c e between c o n d u c t o r s i and k (metres) m = • j 0) y / p - 3 8 -p = e a r t h r e s i s t i v i t y i n Q m These- e q u a t i o n s a r e v a l i d f o r t h e r a n g e |mr|< 0.25 f o r s e l f impedance and |mDik|<0.25 f o r mutual impedance. F o r t h e range |m Dik|>0.25 Wedepohl & W i l c o x s u g g e s t the i n t e g r a t i o n of C O Z i k = j wy 1 /2 2~ - 1 ' /~2 2 va. +m + e / ct +m I I /~2 2~ „ /2 2 | a| + / a +m 2 /a +m 1* / 2, 2 i x . - e / a +m e J dx 2 / V where: (4.6) x' = h o r i z o n t a l d i s t a n c e between c o n d u c t o r i and k 1* = modulus o f t h e d i f f e r e n c e o f t h e d e p t h o f b u r i a l of c o n d u c t o r i and k 4.2 R e p r e s e n t i n g the Ground as One U n d i v i d e d Conductor T h i s m o d e l was s e l e c t e d i n t h i s t h e s i s . I t i n v o l v e s t h e i n t r o d u c t i o n o f a f i c t i t i o u s r e t u r n p a t h w i t h r e s p e c t t o which the i n d u c t a n c e s are c a l c u l a t e d . The ground i s c o n s i d e r e d as one u n d i v i d e d c o n d u c t o r and t h e m u t u a l impedance between t h e g r o u n d and t h e s u b c o n d u c t o r s a r e c a l c u l a t e d a s shown b e l o w . W i t h t h i s a p p r o a c h , eddy c u r r e n t s w h i c h would c i r c u l a t e i n the ground i f c u r r e n t f l o w s i n t o c o n d u c t o r 1 above ground and r e t u r n s t h r o u g h c o n d u c t o r 2 a b o v e g r o u n d , a r e i g n o r e d . In r e f e r e n c e [ 1 0 ] i t has been shown t h a t t h i s e f f e c t i s n e g l i g i b l e up t o 1 kHz f o r t h e c a s e o f a 500 kV o v e r h e a d l i n e . I n t h e l o w e r f r e q u e n c y r e g i o n , t h i s a p p r o a c h g i v e s v e r y a c c u r a t e -39-r e s u l t s , b u t a t h i g h e r f r e q u e n c i e s t h e r e s u l t s m u s t be i n t e r p r e t e d w i t h some c a u t i o n , u n l e s s i t can be shown t h a t t h e s k i n e f f e c t i n t h e c o n d u c t o r s i s much more p r o n o u n c e d t h a n eddy c u r r e n t e f f e c t s i n the ground. The main advantage of t h i s model i s t h a t , b e c a u s e o n l y one c o n d u c t o r i s c o n s i d e r e d f o r the e a r t h , the more c o m p l i c a t e d ground r e t u r n f o r m u l a e need o n l y be used i n one row and one column of the m a t r i x i n e q u a t i o n (3.3). I n a d d i t i o n , t h e r e i s f r e e d o m a s t o t h e c h o i c e o f l o c a t i o n of t h e f i c t i t i o u s r e t u r n p a t h . The d i s t a n c e t o be used i n e q u a t i o n ( 4 . 4 ) can be n e a r l y h a l v e d by c e n t r a l l y l o c a t i n g t h i s p a t h . T h i s r e d u c e s t h e v a l u e s o f t h e p a r a m e t e r )m D i k | i n e q u a t i o n ( 4 . 5 ) , t h e r e b y g i v i n g more a c c u r a t e a n s w e r s and a l s o r e q u i r e s t h e use o f t h e more c o m p l i c a t e d f o r m u l a ( 4 . 6 ) f o r much h i g h e r f r e q u e n c i e s o n l y . T h e l o o p e q u a t i o n s w i t h t h e s o i l i n c l u d e d a s an a d d i t i o n a l c o n d u c t o r would be : ' 1 1 Z, Z, In l g Z, , L g l Z Z z n n z n g gn gg" g J ( 4 . 7 ) i n w h i c h Z. r e f e r s t o t h e m u t u a l i m p e d a n c e b e t w e e n i g s u b c o n d u c t o r s " i " and g r o u n d w i t h common r e t u r n i n " q " . The e q u a t i o n s f o r ground r e t u r n a re v a l i d o n l y when t h e common r e t u r n i s t h e g r o u n d . The n e x t s e c t i o n shows how i s d e r i v e d f o r -40-t h e c a s e where t h e common r e t u r n i s n o t g r o u n d b u t a s e c o n d c o n d u c t o r . 4.2.1 Mutual impedance between a subconductor and ground w i t h common r e t u r n i n another s u b c o n d u c t o r . T h e a n a l y s i s i s d o n e by c o m p a r i n g t w o c i r c u i t s . The f i r s t c i r c u i t r e p r e s e n t s two c o n d u c t o r s w i t h common ground r e t u r n ( F i g u r e 4 . 2 ) . T h i s s i t u a t i o n can be s o l v e d u s i n g the ground r e t u r n f o r m u l a e . The second c i r c u i t r e p r e s e n t s one c o n d u c t o r and t h e g r o u n d , w i t h common r e t u r n i n a s e c o n d c o n d u c t o r ( F i g u r e s 4.3) I t i s t h i s s i t u a t i o n f o r w h i c h a s o l u t i o n i s r e q u i r e d . V l _ L (ground V2<~) r e t u r n ) l i + l a F i g 4,2 Two c o n d u c t o r s w i t h common ground r e t u r n ( g r n n n d ) — ^ _ J _ 4 ! l + *| 2 ( r e t u r n ) F i g 4.3 C i r c u i t of one c o n d u c t o r and the ground w i t h common r e t u r n i n a second c o n d u c t o r . For c i r c u i t #2 the l o o p e q u a t i o n s may be w r i t t e n a s : a u V b - * ^112 L z g 1 2 'lg2 :gg2 J (4.8) w h e r e t h e t h i r d s u b s c r i p t d e n o t e t h e common r e t u r n . The i m p e d a n c e w h i c h we w a n t t o o b t a i n i s f o r m u l a f o r : Z^g2 = Z g l 2 ^ n e c i r c u i t s of F i g u r e s 4.2 and 4.3 are -41-e q u i v a l e n t i f ; V a " V l " V 2 V b - " V 2 I 2 - - d g + I x ) (4.9) F o r c i r c u i t J l t h e l o o p i m p e d a n c e may be w r i t t e n a s : " v l " - V 2 -' l l g :21g '12g !22g-» S u b s t i t u t i n g c o n d i t i o n s (4.8) i n t o e q u a t i o n (4.10) g i v e s : (4.10) V -V V I V2 -V, 2 -J "-"21g Z l l g " Z 2 1 g Z 1 2 g " Z 2 2 8 -Z. -Z 22g - I2-Z l l g + Z 2 2 g + 2 Z 1 2 g Z 2 2 g Z 1 2 g i _ 'bJ u Z 2 2 g _ Z 1 2 g '22g ^ I 8" (4.11) By c o m p a r i n g e q u a t i o n ( 4 . 8 ) and ( 4 . 1 1 ) i t i s p o s s i b l e to c o n c l u d e t h a t : Z l g 2 = Z 2 2 g _ Z 1 2 g where Z 0 0 and Z n o are c a l c u l a t e d u s i n g the ground r e t u r n 22g 12g formulae o f P o l l a c z e k w i t h e q u a t i o n s (4.4) and ( 4 . 5 ) . -42-5 . P I P E T Y P E C A B L E S C a b l e s a r e s o m e t i m e s e n c l o s e d i n p i p e s w h i c h a c t a s d u c t s f o r c o o l i n g l i q u i d s , i n s u l a t i o n g a s e s o r a s p r o t e c t i o n a g a i n s t m e c h a n i c a l o r c h e m i c a l d a m a g e . T h e p i p e m a t e r i a l m a y e i t h e r b e p l a s t i c , a l u m i n i u m o r s t e e l . P l a s t i c a n d a l u m i n i u m p i p e s p o s e n o s p e c i a l p r o b l e m s i n i m p e d a n c e c a l c u l a t i o n s , i n c o n t r a s t t o s t e e l p i p e s w h i c h a r e h i g h l y n o n l i n e a r a s f a r a s t h e i r m a g n e t i c p r o p e r t i e s a r e c o n c e r n e d . B e c a u s e o f t h e a d v a n t a g e s w h i c h s t e e l p o s s e s s e s w i t h r e s p e c t t o s t r e n g t h a n d c o s t , p i p e s a r e u s u a l l y m a d e o f s t e e l u p t o n o w . T h e t r a n s m i s s i o n o f a l t e r n a t i n g c u r r e n t i n p i p e t y p e c a b l e s p r o d u c e s e n e r g y l o s s e s i n t h e p i p e i t s e l f . T h e s e l o s s e s i n t h e p i p e a r e c o m p a r a b l e t o t h e l o s s e s d u e t o t h e s k i n a n d p r o x i m i t y e f f e c t s i n t h e c o n d u c t o r s , a n d i n m a n y c a s e s a r e l a r g e r t h a n t h e l o s s e s c a u s e d b y a l l t h e o t h e r a . c . e f f e c t s c o m b i n e d . T h e y a r i s e f r o m e d d y c u r r e n t s i n d u c e d i n t h e p i p e b y t h e e l e c t r o m a g n e t i c f i e l d c r e a t e d a b y t h e a l t e r n a t i n g c u r r e n t s i n t h e c o n d u c t o r s . I t i s n o t p o s s i b l e t o h a v e c o n s t a n t i m p e d a n c e s f o r s t e e l p i p e , b e c a u s e t h e v a l u e o f t h e i m p e d a n c e i s a f u n c t i o n o f t h e c u r r e n t , d u e t o t h e n o n l i n e a r i t y o f t h e m a t e r i a l . I n o t h e r w o r d s , i n t h e e q u a t i o n B - M u H o r u r w i l l b e a f u n c t i o n o f H . F i g u r e 5 . 1 s h o w s h o w u r v a r i e s w i t h I f o r a t y p i c a l s t e e l p i p e . -43-F i g . 5.1 V a r i a t i o n of r e l a t i v e p e r m e a b i l i t y with c u r r e n t through the p i p e . T h i s f u n c t i o n i s g e n e r a l l y n o n l i n e a r . I n c l u d i n g such a f u n c t i o n i n t o Maxwell's e q u a t i o n s makes t h e i r s o l u t i o n v i r t u a l l y i m p o s s i b l e , even f o r simpler problems.The value of U r >however changes t y p i c a l l y by an order of magnitude as the f i e l d H v a r i e s i n the p i p e . C h o o s i n g an a p p r o p r i a t e " a v e r a g e " or " e f f e c t i v e " p e r m e a b i l i t y v a l u e to be used i n the c a l c u l a t i o n of l o s s e s has t h e r e f o r e been d i f f i c u l t . In p r a c t i c e i s u s u a l l y t r e a t e d as an unknown parameter, which i s chosen to o b t a i n a best f i t to the experimental data. In t h i s s e c t i o n , the n o n l i n e a r i t y i s not t r e a t e d d i r e c t l y i n o r d e r to o b t a i n a c o n s t a n t [Z] m a t r i x . In t h i s procedure the w h o l e s t e e l p i p e i s assumed t o be l i n e a r w i t h a c o n s t a n t p e r m e a b i l i t y , but d i f f e r e n t v a l u e s of p e r m e a b i l i t y are used i n order to study i t s i n f l u e n c e on the [Z] matrix v a l u e s . For the pipe type c a b l e two shapes of f i l a m e n t s are used, namely the c i r c u l a r and " e l e m e n t a l " type. The p i p e l o s s e s a r e a v a i l a b l e f o r d i f f e r e n t c o n d u c t o r c o n f i g u r a t i o n s . P r e v i o u s r e s e a r c h i n t h i s a r e a has proved t h a t t h e c l o s e t r i a n g u l a r c o n f i g u r a t i o n ( s e e F i g u r e 7.11) c a u s e s -44-s m a l l e r e d d y c u r r e n t l o s s e s i n t h e p i p e t h a n t h e c r a d l e c o n f i g u r a t i o n ( s e e F i g u r e 7 . 1 0 ) , b u t i t has t h e w o r s t c o o l i n g p r o p e r t i e s . The open t r i a n g u l a r c o n f i g u r a t i o n (see F i g u r e 7.12), has t h e b e s t c o o l i n g p r o p e r t i e s , but i t i s n o t known w h e t h e r i t has s m a l l e r or b i g g e r l o s s e s than t h o s e of t h e c l o s e d t r i a n g u l a r c o n f i g u r a t i o n s . The l o s s e s a r e s m a l l e r than those f o r the c r a d l e c o n f i g u r a t i o n , because s k i n e f f e c t between c o n d u c t o r s i s s m a l l e r . The o p t i m a l c o n f i g u r a t i o n f o r s p e c i f i e d c u r r e n t r a t i n g cannot be chosen on the b a s i s of e l e c t r i c a l p r o p e r t i e s a l o n e , but r e q u i r e s proper heat t r a n s f e r e v a l u a t i o n s as w e l l . F o r p i p e t y p e c a b l e s , two o p t i o n s have been implemented: a) R e t u r n t h r o u g h p i p e o n l y : I n t h i s c a s e no ground r e t u r n f o r m u l a e are u s e d . The r e t u r n i s c o n s i d e r e d t o be o n l y t h r o u g h the- p i p e . T h i s s i t u a t i o n a r i s e s when t h e c a b l e i s l a i d i n c o n c r e t e d u c t s . The impedance m a t r i x i s a 3x3 m a t r i x i n t h i s c a s e . b) R e t u r n t h r o u g h p i p e and ground: Ground r e t u r n f o r m u l a e a r e u s e d a n d ' t h e i m p e d a n c e m a t r i x i s a 4x4 m a t r i x (3 c o n d u c t o r s + p i p e ) S k i n and p r o x i m i t y e f f e c t s a r e s t u d i e d f o r t h e c a s e of a t h r e e - p h a s e c a b l e s y s t e m e n c l o s e d i n a p i p e . F o r t h i s , t h e c u r r e n t d i s t r i b u t i o n t h r o u g h the p i p e and c o n d u c t o r s i s o b t a i n e d . I t i s an i n t e r e s t i n g f a c t , not n o r m a l l y r e c o g n i z e d , t h a t t h e sum o f a l l f i l a m e n t c u r r e n t a m p l i t u d e s i s g r e a t e r t h a n t h e t o t a l c u r r e n t w h i c h w o u l d be measured by a c u r r e n t m e t e r p l a c e d i n s e r i e s w i t h t h e t r a n s m i s s i o n l i n e . T h i s f a c t w i l l be more r e a d i l y u n d e r s t o o d by examining the phasor diagram of F i g u r e 5.3 . -45-A c c o r d i n g t o G r a n e a u , " t h e t o t a l c u r r e n t c a n be e x p r e s s e d by the f o l l o w i n g phasor sum I r = -OA + AO - j OB +j OB + OD (5.1) F i g . 5.2 P h a s o r d i a g r a m f o r one c o n d u c t o r o f a t h r e e c o r e c a b l e . The u s e f u l f i l a m e n t c u r r e n t components t h a t a c t i n t h e d i r e c t i o n f r o m 0 t o D a r e n o t u n i f o r m l y d i s t r i b u t e d o v e r t h e c o n d u c t o r c r o s s s e c t i o n . F o r t h i s r e a s o n a l o n e t h e a . c . r e s i s t a n c e o f t h e c o n d u c t o r w o u l d be g r e a t e r t h a n i t s d . c . r e s i s t a n c e . However, t h i s i s n o t t h e f u l l e x p l a n a t i o n when the a.c. l o s s e s c o n t r i b u t i o n i s v e r y l a r g e . There may e x i s t , as seen i n F i g u r e 5.2 two more s e t s o f c l o s e d c u r r e n t s OA and AO as w e l l as BO and OB, w h i c h do n o t l e a v e t h e c o n d u c t o r . These s e t s o f i n t e r n a l c i r c u l a t i n g c u r r e n t a l s o p r o d u c e j o u l e heat as a r e s u l t o f e l e c t r o n s c a t t e r i n g w h i c h has t o be added t o t h e l o s s power t h a t c a n be d i r e c t l y a t t r i b u t e d t o t h e n o n - c i r c u l a t i n g u s e f u l c u r r e n t . The i n t e r n a l c i r c u l a t i n g c u r r e n t f l o w s a l o n g t h e c o n d u c t o r s i n one s e t o f f i l a m e n t s a n d a r e r e t u r n e d by t h e r e m a i n i n g f i l a m e n t s . At t h e ends o f t h e c o n d u c t o r s t h e r e must e x i s t s t r a n s v e r s e c u r r e n t s which connect the go and r e t u r n s t r e a m s " . - 4 6 -6 . S T R A N D E D C O N D U C T O R S I n a l l d i s c u s s i o n s s o f a r i t h a s b e e n a s s u m e d t h a t a l l t h e c o n d u c t o r s a r e s o l i d r o u n d o r t u b e s i n w h i c h t h e c u r r e n t f l o w s p a r a l l e l t o t h e c o n d u c t o r a x i s . T h i s i s n o t n e c e s s a r i l y t r u e f o r c o n d u c t o r s m a d e u p o f w i r e s t w i s t e d t o g e t h e r . T h e p r i m a r y p u r p o s e o f t h e s t r a n d e d c o n s t r u c t i o n i s t o m a k e t h e c o n d u c t o r s u f f i c i e n t l y f l e x i b l e f o r t h e s t o r a g e o f l o n g l e n g t h s o f c a b l e o n r e e l s . M a n y e x p e r i m e n t s w e r e m a d e d u r i n g t h e y e a r s 1 9 1 5 t o 1 9 6 8 a b o u t t h e e f f e c t o f s p i r a l l i n g o n s k i n a n d p r o x i m i t y e f f e c t s , w i t h s o m e c o n t r a d i c t o r y r e s u l t s . T o d a y t h e r e r e m a i n s n o d o u b t t h a t i n s t r a n d e d a l u m i n i u m c o n d u c t o r s , t h e c u r r e n t f o l l o w s t h e h e l i c a l p a t h o f i n d i v i d u a l w i r e s b e c a u s e t h e t o u g h o x i d e c o a t i n g p r e v e n t s i n t e r w i r e c o n d u c t i n g . O n e c o n s e q u e n c e o f t h e h e l i c a l c u r r e n t p a t t e r n i s t h e l e n g t h e n i n g o f t h e c u r r e n t p a t h a n d t h e a t t e n d a n t i n c r e a s e i n d . c . r e s i s t a n c e . T o c o m p u t e t h i s r e s i s t a n c e i t i s c u s t o m a r y t o u s e t h e a c t u a l c o n d u c t o r l e n g t h , w h i c h i g n o r e s t h e h e l i c a l p a t h e x t e n s i o n , a n d t h e n o r m a l c r o s s - s e c t i o n a l a r e a o f t h e w i r e s a s i f t h e y w e r e n o t l a i d u p i n h e l i c e s , a n d t h e n m u l t i p l y t h e r e s i s t a n c e b y a s t r a n d i n g f a c t o r w h i c h d e p e n d s o n t h e l e n g t h a n d t h e r e f o r e t h e " s i z e " o f t h e c o n d u c t o r s . T h i s f a c t o r i s s h o w n i n T a b l e 6 . 1 T a b l e 6 . 1  S t r a n d i n g F a c t o r # o f w i r e s i n c o n d u c t o r 7 1 9 3 7 6 1 9 1 1 2 7 S t r a n d i n g F a c t o r 1 . 1 4 5 1 . 0 5 9 1 . 0 2 8 1 . 0 1 7 1 . 0 1 1 1 . 0 0 8 - 4 7 -A l l t h a t c a n b e s a i d w i t h c e r t a i n t y a b o u t t h e a . c . r e s i s t a n c e o f a c o n c e n t r i c a l l y s t r a n d e d c o n d u c t o r i s t h a t i t w i l l b e g r e a t e r t h a n t h e d . c . r e s i s t a n c e . T h e o x i d e l a y e r s o n t h e w i r e s u r f a c e s d o n o t i n t e r f e r e w i t h t h e s k i n e f f e c t f o r m a t i o n p r o c e s s i n t h e c o n d u c t o r a s a w h o l e . T h e f i n i t e e l e m e n t a n a l y s i s w o u l d s e e m t o b e i d e a l l y s u i t e d f o r c o m p u t i n g t h e a . c . r e s i s t a n c e o f a s t r a n d e d c o n d u c t o r , b u t t h e l a c k o f a s a t i s f a c t o r y m u t u a l i n d u c t a n c e f o r m u l a f o r p a r a l l e l h e l i c e s m a k e s t h i s a p p r o a c h i m p r a c t i c a b l e . T h e r e f o r e , t h e r e s i s t a n c e a n d s e l f i n d u c t a n c e o f a s t r a n d e d c o n d u c t o r w i l l b e c a l c u l a t e d h e r e w i t h p a r a l l e l f i l a m e n t s . G a l l o w a y e t a l [ 2 0 ] d e v e l o p e d a f o r m u l a f o r t h e i n t e r n a l i m p e d a n c e o f a s t r a n d e d c o n d u c t o r , w h i c h i s R c = X c = 2 . 2 5 / U) U 0 U r P ( 6 . 1 ) r TT ( 2 + n ) 1 . 4 1 4 2 w i t h : y r = r e l a t i v e p e r m e a b i l i t y U Q= 4 TT 1 0 " 7 ( H / m ) w = a n g u l a r f r e q u e n c y ( r a d / s ) p = c o n d u c t o r r e s i s t i v i t y ( ft m ) r = r a d i u s o f e a c h o u t e r s t r a n d ( m ) n = n u m b e r o f o u t e r s t r a n d s T h i s f o r m u l a w a s d e v e l o p e d f o r f r e q u e n c i e s a b o v e a f e w k H z , w h e r e i t c a n b e a s s u m e d t h a t t h e c u r r e n t i s c o n f i n e d t o t h e s u r f a c e o f t h e o u t e r l a y e r o f t h e s t r a n d s , b e c a u s e o f s k i n e f f e c t . B y a s s u m i n g t h a t t h e d e p t h o f p e n e t r a t i o n i s v e r y s m a l l , t h e c u r r e n t d e n s i t y o n t h e s u r f a c e o f t h e c o n d u c t o r i s -48-p r o p o r t i o n a l t o the m a g n e t i c - f i e l d i n t e n s i t y on t h e s u r f a c e . The f a c t o r 2.25 was f o u n d e x p e r i m e n t a l l y from f i e l d p l o t t i n g i n an e l e c t r o l y t i c t a n k . A c c o r d i n g t o t h e a u t h o r s , t h i s f o r m u l a g i v e s r e a s o n a b l y a c c u r a t e r e s u l t s a t f r e q u e n c i e s above 2 t o 5 kHz f o r t h e most commonly u s e d s t r a n d e d c o n d u c t o r s w i t h t h e number of o u t e r s t r a n d s e i t h e r b e i n g 6,12,18, or 24. The s t r a n d e d c o n d u c t o r o f F i g u r e 6.1 was s i m u l a t e d by s u b d i v i d i n g each s u b c o n d u c t o r i n t o c i r c u l a r f i l a m e n t s . Because t h e i m p e d a n c e was t o be c a l c u l a t e d f o r h i g h f r e q u e n c i e s , t h e f o l l o w i n g s i m p l i f i c a t i o n s c o u l d be made: F i g . 6.1 Stranded c o n d u c t o r . a) At h i g h f r e q u e n c i e s t h e c u r r e n t f l o w s o n l y i n t h e o u t e r l a y e r o f c o n d u c t o r s due t o t h e p r o x i m i t y e f f e c t . Because of t h i s , o n l y t h e o u t e r l a y e r o f s t r a n d s w i l l be c o n s i d e r e d , s i n c e t h e i n n e r s t r a n d s c a r r y p r a c t i c a l l y no c u r r e n t . b) Symmetry i s e x p l o i t e d . A l l the c o n d u c t o r s i n t h e o u t e r l a y e r are s u b j e c t t o the same c o n d i t i o n s , so t h a t each con d u c t o r w i l l have the same c u r r e n t d i s t r i b u t i o n . T h i s l e a d s t o the f o l l o w i n g s i m p l i f i c a t i o n i n t h e [Z] m a t r i x : I n i t i a l l y t h e s y s t e m o f e q u a t i o n s f o r a s t r a n d e d -49-c o n d u c t o r w i t h N c o n d u c t o r s i n the o u t e r l a y e r i s : AV. AV; AV: AV AV N r -1 Z l l 512 ^21 ^22 Z N I 32 J r2 Z 1 N Z l r ^2N ^ 2 r Z N N z N r rN r r u. r (6.2) whereAV r, I a r e the v o l t a g e drop and the c u r r e n t t h r o u g h t h e r e t u r n c o n d u c t o r . T h i s r e t u r n c o n d u c t o r i s s i m u l a t e d as an u n d i v i d e d c o n d u c t o r l o c a t e d v e r y f a r away f r o m t h e s t r a n d e d c o n d u c t o r i n o r d e r t o a v o i d a n y a l t e r a t i o n i n t h e c u r r e n t d i s t r i b u t i o n due t o p r o x i m i t y e f f e c t . I f we c o n s i d e r t h a t each c o n d u c t o r i s s u b d i v i d e d i n t o "n" c i r c u l a r f i l a m e n t s then : V. = l AV AV i l i 2 AV. •— i n -* I . = l : i l i 2 _ I . ^ *- i n J But: Z. . Z i l j l • • • • Z i l j n ' i n j l ' * A V 1 = A V 2 " A V 3 -and because of symmetry c o n d i t i o n s h - X 2 - X 3 Z. . . m j n - 1 AV N •N (6.3) So the f i r s t and l a s t rows of e q u a t i o n (6.2) become: A V 1 - Z l l X l + Z 1 2 12 + ••• Z1H h + Z l r I r AV " Z r l J r l + Z r 2 X 2 + • • • z r N + Z I r r r - 5 0 -AV X = [ Z 11 + Z 1 2 + z 1 N ] I + Z l r I r AV r - [ Z r l + Z r 2 +, Z „ ] I + Z I r rN J r r T h e r e f o r e , t h e f u l l s y s t e m o f e q u a t i o n s ( 6 . 2 ) r e d u c e s t o t h e f o l l o w i n g reduced system " A V j -AV i _ r "11 L - Z r l ! ' l r r r -1 I I '11 ~ Z l l + Z 1 2 + * * , Z 1 N l r r l r2 rN (6.4) (6.5) w i t h an u n s y m m e t r i c a l m a t r i x . The new m a t r i x [Z ] i s r e d u c e d by G a u s s i a n e l i m i n a t i o n , and f i n a l l y a two by two m a t r i x i s o b t a i n e d : " AV~ 11 l r r r (6.6) w i t h I = t o t a l c u r r e n t t h r o u g h one o f t h e s u b c o n d u c t o r s of the o u t e r l a y e r Now t h e c o n d i t i o n o f z e r o c u r r e n t t h r o u g h t h e r e t u r n p a t h i s i n t r o d u c e d . T h i s c o n d i t i o n i s : N * I + I = 0 I = - I /N r AV = - h l I r / N + C l r I r AV r = - C r l I r / N + I r AV X - AV r -( - C n / N + C r l/N + C l r - C r r ) I r -51-AV = C - C n / N - C r l / » - C l r + C r r C o m p a r i n g t h i s e q u a t i o n w i t h e q u a t i o n ( 3 . 1 4 ) i t i s l o g i c a l t o expect t h a t ? r l / N - 5 l r w h i c h was t r u e i n a l l t h e s t u d i e s d o n e f o r t h e c a s e o f s t r a n d e d c o n d u c t o r s . £ i s t h e impedance o f t h e l o o p f o r m e d by t h e s t r a n d e d c o n d u c t o r and t h e r e t u r n c o n d u c t o r , b u t t h e o b j e c t i v e i s t o c a l c u l a t e t h e r e s i s t a n c e and s e l f i n d u c t a n c e o f t h e s t r a n d e d c o n d u c t o r a l o n e . I t i s known t h a t : Rloop = R s t r a n d e d + R r e t u r n c o n d u c t o r c o n d u c t o r L l o o p = L e x t e r n a l + L i n t e r n a l + L i n t e r n a l s t r a n d e d r e t u r n c o n d u c t o r conductor R r e t u r n = p °>/m condu c t o r TT r 2 r r = r a d i u s of r e t u r n c o n d u c t o r (m) p^ = r e s i s t i v i t y of r e t u r n c o n d u c t o r ( Q m) R s t r a n d e d = Rloop - p con d u c t o r Trr 2 r -52-L i n t e r n a l = W 0.5 r e t u r n c o n d u c t o r 4 TT For t h e c a l c u l a t i o n of L e x t e r n a l i t i s assumed t h a t t h e s t r a n d e d c o n d u c t o r can be s u b s t i t u t e d by a s o l i d round c o n d u c t o r , because the r e t u r n c o n d u c t o r i s l o c a t e d f a r away. L e x t e r n a l = 1 4 In D D D = d i s t a n c e between s t r a n d e d and r e t u r n c o n d u c t o r r = r a d i u s of r e t u r n c o n d u c t o r r r g = r a d i u s of e q u i v a l e n t h o l e c o n d u c t o r - 7 2 Z i n t e r n a l = Z l o o p - 2 wlO I n D ~ ~ -0.25 r r e s r I n t h i s way t h e r e s i s t a n c e and s e l f i n d u c t a n c e o f t h e s t r a n d e d c o n d u c t o r can be d e t e r m i n e d . -53 -7. RESULTS 7 . 1 C o m p a r i s o n o f t h e M e t h o d o f S u b d i v i s i o n w i t h S t a n d a r d  Methods T h i s s e c t i o n shows how s k i n and p r o x i m i t y e f f e c t s a r e t a k e n i n t o a c c o u n t by s u b d i v i d i n g t h e c o n d u c t o r s i n t o f i l a m e n t s . The impedance of a c o a x i a l c a b l e i s c a l c u l a t e d by the f o l l o w i n g methods: a) UBC/BPA l i n e c o n s t a n t s p r o g r a m ( s t a n d a r d m e t h o d ) w h i c h g i v e s an e x a c t s o l u t i o n o f t h e i m p e d a n c e w i t h t h e s k i n e f f e c t f o r m u l a b a s e d on B e s s e l and K e l v i n f u n c t i o n s . b) M e t h o d o f s u b d i v i s i o n u s i n g c i r c u l a r c r o s s - s e c t i o n f i l a m e n t s . c ) M e t h o d o f s u b d i v i s i o n u s i n g s q u a r e c r o s s - s e c t i o n f i l a m e n t s . d) Method o f s u b d i v i s i o n u s i n g " e l e m e n t a l " c r o s s - s e c t i o n f i l a m e n t s . The c o a x i a l c a b l e under study i s shown i n F i g u r e 7.1 : F i g . 7.1 C o a x i a l c a b l e . \ -54-CABLE CHARACTERISTICS -Conductor r a d i u s -Sheath i n n e r r a d i u s -Sheath o u t e r r a d i u s - R e l a t i v e p e r m i t t i v i t y of main i n s u l a t i o n . - R e l a t i v e p e r m i t t i v i t y of c a b l e c o v e r i n g - R e s i s t i v i t y of copper co n d u c t o r - R e s i s t i v i t y of l e a d sheath - R e s i s t i v i t y of s o i l - C able l e n g t h -Average depth of l a y i n g r , = r„ = r „ = 0.0150 m 0.0250 m 0.0300 m 3.5 c2 = 8.0 v-8 p = 1.7 10 fim cu p = 2.1 1 0 ~ 7 fim s p = 50 fim e % = 633 m d - 1.2 m The r e f e r e n c e v a l u e s a r e t h o s e o b t a i n e d u s i n g t h e UBC/BPA program. I n o r d e r t o check f o r s k i n e f f e c t , t h e impedance of the c o a x i a l c a b l e i s c a l c u l a t e d f o r d i f f e r e n t f r e q u e n c i e s . T a b l e s 7.1 and 7.2 show t h e v a r i a t i o n of t h e r e s i s t a n c e and i n d u c t a n c e w i t h f r e q u e n c y f o r e a c h method, as w e l l as t h e p e r c e n t a g e o f e r r o r w i t h r e s p e c t t o UBC/BPA r e s u l t s . F i g u r e s 7.2 and 7.3 a r e t h e c o r r e s p o n d i n g p l o t s . -55-TABLE 7.1 V a r i a t i o n of R e s i s t a n c e w i t h Frequency f o r the Four D i f f e r e n t Methods Frequency Hz UBC Program ft/km C i r c l e s ft/km % Squares ft/km % E l e m e n t a l ft/km % 0.1 0.26711109 0.320025 19 0.261250 2.1 0.2671231 0.0 1.0 0.26711245 0.320026 19 0.261251 2.1 0.2671244 0.0 10.0 0.26724798 0.320143 19 0.261385 2.1 0.2672563 0.0 100.0 0.27716327 0.328418 18 0.270433 2.4 0.2773971 0.0 1000.0 0.34337618 0.362009 5 0.308502 10 0.3492723 1.7 10000.0 0.81908315 0.369224 54 0.323056 60 0.8357925 2.0 TABLE 7.2 V a r i a t i o n of Indu c t a n c e w i t h Frequency f o r the Four D i f f e r e n t Methods Frequency Hz UBC Program (mH/km) C i r c l e s (mH/km) % Squares (mH/km) % E l e m e n t a l (mH/km) % 0.1 0.16729715 0.14769579 11 0.151737 9 0.1555724 7 1.0 0.16729715 0.14761621 11 0.151736 9 0.1555713 7 10.0 0.16716806 0.1475048 11 0.151589 9 0.1554589 7 100.0 0.157944497 0.13921315 12 0.141885 10 0.1470217 10 1000.0 0.129258941 0.11682456 9 0.118398 8 0.1234355 5 10000.0 0.115896709 0.11459887 1 0.115205 1 0.1128431 2 50000.0 0.108246983 0.11456928 6 0.115207 6 0.1102921 2 -56-1 . 70 1 60 1 .50 -1 .40 -1 30 1.20 J . 10 -J 1 .00 ~ 0.90 | 0 . 6 0 : £ 0.70-~ 0.60 : LU z 0.50 cr (T! 0 . 4Q -•—. £ 0.30 0 20 0 '0 -— UBC Program + E l e m e n t a l x C i r c l e s A Squares -0 .00 \ i i ••|... .| i . . i|i .| i V v y n i • ii|i.ini»nnnnnn^ , i i ' i""i""i 'vrvw 1 • 1 •I""|""I'1'<iVTTI 10"' 2 3 4 6 1 2 3 4 6 10 2 3 4 6 10' 2 3 4 6 10' FREQUENCY I HZ ) 2 3 4 6 10* 2 3 4 C 10 FIG 7.2 V a r i a t i o n of R e s i s t a n c e w i t h Frequency f o r the Four D i f f e r e n t Methods . 0.30 0.28 0.26 0.24 0.22 -0.20 -0.18 : -0.00 4 — UBC Program + E l e m e n t a l x C i r c l e s A Squares -i T^'i'i'w ' "•i""i"'q|i'yrwi 1 • "i""i""i'VvvrTi •1 ' - , 10" 2 3 4 6 1 2 3 4 6 10 2 3 4 6 ltf 2 3 4 6 10' 2 3 4 6 10 2 3 4 6 10 FREQUENCY ( Hi ) FIG 7.3 V a r i a t i o n of In d u c t a n c e w i t h Fre.quency f o r the Four D i f f e r e n t Methods. 'I"I"I"1'1'VH1 -57-Th e same number of f i l a m e n t s was used f o r each type of s u b d i v i s i o n . As can be seen from T a b l e 7.1 and T a b l e 7.2, t h e r e s u l t s o b t a i n e d t h r o u g h t h e s u b d i v i s i o n method a r e much b e t t e r a t the l o w e r f r e q u e n c i e s ( l e s s t h a n 1 0 0 0 0 H z f o r t h i s n u m b e r o f f i l a m e n t s ) and t h e y a r e a s y m p t o t i c to a v a l u e w h i c h we w i l l c a l l t h e C u t - O f f - R e s i s t a n c e (COR). The COR i s a c h a r a c t e r i s t i c f o r t h i s t y p e o f p r o c e d u r e and s h o u l d be e q u a l to the d.c, r e s i s t a n c e of t h e o u t e r f i l a m e n t s of t h e c o n d u c t o r s . The c u t o f f r e s i s t a n c e can be e x p l a i n e d as f o l l o w s : As t h e f r e q u e n c y i n c r e a s e s , more and more c u r r e n t t e n d s t o f l o w i n t h e o u t e r f i l a m e n t s o f t h e i n n e r c o n d u c t o r and i n t h e i n n e r f i l a m e n t s of the o u t e r c o n d u c t o r s . At h i g h f r e q u e n c i e s v i r t u a l l y a l l t h e c u r r e n t i s f l o w i n g t h r o u g h them, t h a t i s , i t c o n s t i t u t e s t h e o n l y p a t h f o r c u r r e n t f l o w and i t s d.c. r e s i s t a n c e i s s e e n as t h e e f f e c t i v e r e s i s t a n c e . T h i s t y p e o f p r o c e d u r e c a n n o t g e n e r a t e s u b s t a n t i a l l y h i g h e r r e s i s t a n c e s because i t assumes t h a t the c u r r e n t d e n s i t y i n each f i l a m e n t i s u n i f o r m . I n g e n e r a l , t h e COR o f any f i n i t e e l e m e n t p r o c e d u r e i s c l o s e l y r e l a t e d t o the d.c. r e s i s t a n c e o f t h e f i l a m e n t s c o v e r i n g t h e c o n d u c t o r a r e a i n which c u r r e n t t e n d s t o c o n c e n t r a t e a t h i g h f r e q u e n c i e s . T h u s , t o i n c r e a s e t h e COR'S and t h e r e b y t h e u s a b l e f r e q u e n c y band o f a s u b d i v i s i o n p r o c e d u r e , one must i n c r e a s e the d . c . r e s i s t a n c e s of t h e s e f i l a m e n t s . In o t h e r words, one must d e c r e a s e t h e i r c r o s s - s e c t i o n a l a r e a a t t h e e x p e n s e o f more comp u t a t i o n time and memory r e q u i r e m e n t s . I n t h i s p a r t i c u l a r s i m u l a t i o n , t h e b e s t r e s u l t s f o r the -58-same number of f i l a m e n t s were o b t a i n e d f o r t h e " e l e m e n t a l " c r o s s s e c t i o n s hape. The r e a s o n i s c l e a r l y t h e f a c t t h a t t h i s t y p e of f i l a m e n t c o v e r s c o m p l e t e l y t h e a r e a of t h e c o a x i a l c a b l e , and t h a t a f i l a m e n t w i d t h was chosen w h i c h decays e x p o n e n t i a l l y w i t h t h e d i s t a n c e t o model s k i n e f f e c t more a c c u r a t e l y . For t h e case of square f i l a m e n t s the e r r o r s a r e a l s o s m a l l i n the lower f r e q u e n c y r a n g e , w i t h t h e r e s i s t a n c e v a l u e s b e i n g somewhat s m a l l e r t h a n t h e e x a c t v a l u e s , s i n c e t h e s u b d i v i s i o n c o v e r s s l i g h t l y more a r e a t h a n t h e t o t a l a r e a of t h e c o a x i a l c a b l e . J u s t t h e o p p o s i t e happens w i t h t h e c i r c u l a r f i l a m e n t s where the a r e a c o v e r e d by t h e f i l a m e n t s i s much l e s s t h a n t h e t o t a l a r e a , so t h a t r e s i s t a n c e s become h i g h e r t h a n t h e e x a c t o n e s . The r e s u l t s o b t a i n e d f o r i n d u c t a n c e and r e s i s t a n c e a r e r e a s o n a b l y g o o d f o r a l l t h r e e t y p e s o f f i l a m e n t s . I f more a c c u r a c y i s r e q u i r e d , e s p e c i a l l y a t h i g h e r f r e q u e n c i e s , t h e n more numbers of f i l a m e n t s a r e r e q u i r e d . I n o r d e r t o c h e c k i f p r o x i m i t y e f f e c t i s t a k e n i n t o a c c o u n t as w e l l , i n a d d i t i o n t o s k i n e f f e c t , l e t us see how t h e i m p e d a n c e c h a n g e s a t a g i v e n f r e q u e n c y ( 1 0 0 0 0 H z ) when t h e d i s t a n c e "D" o f t h e n o n c o n c e n t r i c a r r a n g e m e n t i n F i g u r e 7.4 i s v a r i e d . The r e s u l t s are shown i n T a b l e 7.3. -59-F i g . 7.4 V a r i a t i o n of the impedance w i t h the p o s i t i o n of the c e n t r a l c o n d u c t o r . TABLE 7.3 V a r i a t i o n of Impedance of a C o a x i a l Cable w i t h P o s i t i o n of C e n t r a l Conductor D (m) R e s i s t a n c e ( Q/km) Reactance ( ft/km) 0. 0.36200 0.73403 0.001 0.36322 0.73246 0.002 0.36688 0.72773 0.003 0.37307 0.71985 0.004 0.38191 0.71985 0.005 0.39362 0.69462 0.006 0.40845 0.67730 0.007 0.42676 0.65690 0.008 0.44898 0.63353 0.009 0.47571 0.60735 0.010 0.50784 0.57864 As we i n c r e a s e the d i s t a n c e "D" the r e s i s t a n c e i n c r e a s e s , because the c u r r e n t d i s t r i b u t i o n i s becoming l e s s u n i f o r m due t o p r o x i m i t y e f f e c t . F o r t h e c a s e o f t h e i n d u c t a n c e , due t o t h e a v e r a g e i n c r e a s e o f t h e d i s t a n c e between t h e f i l a m e n t s o f t h e - 6 0 -c e n t r a l c o n d u c t o r and t h o s e of t h e s h e a t h , i t d e c r e a s e s w i t h an i n c r e a s e i n the d i s t a n c e "D". - 6 1 -7.2 Comparison of Ground R e t u r n Formulae The v a r i a t i o n of ground r e t u r n impedance w i t h f r e q u e n c y , as g i v e n by some o f t h e f o r m u l a e , a r e compared i n t h i s s e c t i o n . A c c o r d i n g t o C a r s o n , t h e v a r i a t i o n of t h e g r o u n d r e t u r n i m p e d a n c e w i t h t h e d e p t h o f b u r i a l o f a c o n d u c t o r i s m i n i m a l , and f o r most f r e q u e n c i e s e q u a t i o n ( 4 . 2 ) can t h e r e f o r e be used. T h i s i s v e r i f i e d , f o r t h e example of t h e c o a x i a l c a b l e , by c a l c u l a t i n g t h e impedance o f a b u r i e d c o n d u c t o r w i t h ground r e t u r n f o r v a r i o u s d e p t h o f b u r i a l . T h e r e s u l t s a r e those of F i g u r e 7.5 and Table 7.4 . TABLE 7.4 V a r i a t i o n of the Impedance of a B u r i e d  Conductor w i t h Depth of B u r i a l Depth of B u r i a l (m) Impedance ( ft /km) -2.0 1.480056271 -1.5 1.480242021 -1.0 1.480428594 -0.5 1.480514709 0. 1.480701324 0.5 1.480887164 1.0 1.480973343 1.5 1.481160022 2.0 1.481345930 -62-£ 1.400 -2 0 -1.1 1.5 <!.0 F i g . 7.5 V a r i a t i o n of the impedance of a b u r i e d c o n d u c t o r w i t h depth of b u r i a l . I n o r d e r t o compare t h e v a r i o u s e a r t h r e t u r n f o r m u l a s , the impedance of a c o a x i a l c a b l e s i m i l a r t o t h a t o f F i g u r e 7.1 ( t h e o n l y d i f f e r e n c e s a r e i n t h e dimensions r^O.0234 m r 2=0.0385 m r 3=0.0431 m ) i s c a l c u l a t e d by u s i n g : a ) W e d e p o h l 1 s a p p r o x i m a t e f o r m u l a w i t h t h e method o f s u b c o n d u c t o r s b) W e d e p o h l ' s a p p r o x i m a t e f o r m u l a w i t h t h e UBC. L i n e Parameters Program. c) P o l l a c z e k ' s f o r m u l a w i t h the BPA EMTP. The d e p t h o f b u r i a l i s two m e t e r s . I n e a c h c a s e a 2x2 m a t r i x i s o b t a i n e d . The r e s u l t s f o r Z^^ ( s e l f impedance of the c e n t r a l c o n d u c t o r w i t h ground r e t u r n ) a r e shown i n T a b l e 7.5 and F i g u r e 7.6, f o r Z 2 2 ( s e l f i mpedance o f t h e s h e a t h w i t h g r o u n d r e t u r n ) i n F i g u r e 7.7 a n d T a b l e 7.6 a n d f i n a l l y f o r Z^ 2 ( m u t u a l i m p e d a n c e b e t w e e n c e n t r a l c o n d u c t o r and s h e a t h i n p r e s e n c e o f e a r t h ) a r e shown i n F i g u r e 7.8 and T a b l e 7.7. A s c a n be s e e n , t h e r e s u l t s f r o m t h e f i r s t t w o - 6 3 -a p p r o x i m a t e methods a r e a l m o s t i d e n t i c a l . The b i g g e s t d i f f e r e n c e i s a p p r o x i m a t e l y o f 2% a t h i g h f r e q u e n c i e s , w h e r e t h e s u b d i v i s i o n can cause p r o b l e m s . The d i f f e r e n c e between t h e s e two m e t h o d s a n d P o l l a c z e k ' s f o r m u l a f o r t h e r e s i s t a n c e s i s n e g l i g i b l e a t l o w f r e q u e n c i e s a n d r e a c h e s 1 7 % a t h i g h f r e q u e n c i e s . T h i s i s to be e x p e c t e d b e c a u s e Wedepohl and W i l c o x s t a t e t h a t t h e i r f o r m u l a s a r e a c c u r a t e up to f r e q u e n c i e s of 1.7 MHz i f t h e s e p a r a t i o n b e t w e e n t h e c o n d u c t o r s i s l e s s t h a n 30cm, o r a p p l i e d t o t h e c a s e s t u d i e d h e r e up t o f r e q u e n c i e s of 160 kHz f o r s e p a r a t i o n s o f a p p r o x i m a t e l y 1.0 m b e t w e e n t h e c o n d u c t o r s . The r e a c t a n c e v a l u e s c a l c u l a t e d w i t h Wedepohl's f o r m u l a s ( m e t h o d s a and b) d i f f e r more f r o m P o l l a c z e k ' s f o r m u l a s t h a n t h e r e s i s t a n c e s . The d e v i a t i o n i s between 8% a t power f r e q u e n c y and 15% a t 0.1 MHz. I t a r i s e s from t h e f a c t t h a t methods a and b ar e a p p r o x i m a t i o n s . F or p r a c t i c a l p u r p o s e s t h e s e d i f f e r e n c e s are n e g l i g i b l y s m a l l , e s p e c i a l l y i f one c o n s i d e r s t h e f a c t t h a t t h e e a r t h r e s i s t i v i t y i s n e i t h e r u n i f o r m nor w e l l known. TABLE 7.5 S e l f Impedance of the C e n t r a l Conductor of a C o a x i a l Cable w i t h Ground R e t u r n FREQUENCY (Hz) RESISTANCE ( ft /km ) REACTANCE ( f t /km) P o l l a c z e k Wedepohl Wedepohl S u b d i v i s i o n P o l l a c z e k Wedepohl Wedepohl S u b d i v i s i o n 0.01 0.00989238 0.00988745 0.0098767 0.000189762 0.00020274 0.0000206 0.10 0.00998122 0.00997633 0.0099655 0.001752957 0.00188271 0.00191 1.00 0.01087214 0.0186853 0.010857 0.01608322 0.01737987 0.017652 10.00 0.02005585 0.0200939 0.020055 0.1463281 0.15926967 0.16199 100.00 0.1184174 0.12049795 0.11798 1.305451 1.43331063 1.4621 1000.00 1.029044 1.0662314 1.0364 11.50433 12.7495429 13.087 10000.00 9.783653 10.6211385 10.564 100.5515 112.219055 115.92 100000.00 94.12958 119.181138 119.78 864.122 964.704343 1000.1 TABLE 7.6 S e l f Impedance of the Sheath of a C o a x i a l Cable w i t h Ground R e t u r n FREQUENCY (Hz) RESISTANCE ( ft /km ) REACTANCE ( ft / km) P o l l a c z e k Wedepohl Wedepohl S u b d i v i s i o n P o l l a c z e k Wedepohl Wedepohl S u b d i v i s i o n 0.01 0.2991733 0.2991733 0.29655 0.000179764 0.00019349 0.00019613 0.10 0.2992621 0.29926215 0.29663 0.001652987 0.00179022 0.0018166 1.00 0.3001497 0.30015105 0.29752 0.01508357 0.01645507 0.016719 10.00 0.3090124 0.30905405 0.30643 0.1363825 0.15006882 0.15271 100.00 0.3972241 0.39852024 0.39590 1.219603 1.35555727 1.3820 1000.00 1.268331 1.30667404 1.3046 10.76320 12.0944344 12.358 10000.00 9.756521 10.7883984 10.832 93.56410 106.035753 108.66 100000.00 92.25025 119.681573 120.05 799.7925 902.604509 927.55 TABLE 7.7 Mutual Impedance Between Sheath and C e n t r a l Conductor i n Presence of E a r t h FREQUENCY (Hz) RESISTANCE ( ft /km ) REACTANCE ( ft / km) P o l l a c z e k Wedepohl Wedepohl S u b d i v i s i o n P o l l a c z e k Wedepohl Wedepohl S u b d i v i s i o n 0.01 0.98689E-05 0.9870E-05 0.98703E-05 0.000179912 0.00019321 0.00019627 0.10 0.98675E-04 0.9870E-04 0.98763E-03 0.001654456 0.00178738 0.0018180 1.00 0.98630E-03 0.9876E-03 0.98763E-03 0.01509826 0.01642668 0.016733 10.00 0.98490E-02 0.9890E-02 0.98907E-02 0.1365294 0.14978483 0.15285 100.00 0.98077E-01 0.9935E-01 0.99362E-01 1.221067 1.35271743 1.3834 1000.00 0.9696714 1.00695299 1.00800 10.77789 12.0660515 12.373 10000.00 9.503526 10.4376690 10.535 93.69664 105.766173 108.80 100000.00 92.31990 118.560659 119.75 799.7987 901.495758 928.96 -67-io3 n 4 3 FJPG 7.6 V a r i a t i o n w i t h Frequency of S e l f Impedance of C e n t r a l .Conductor i n a C o a x i a l C a b l e . -68-io3 , 4-3^ F I G 7 . 7 V a r i a t i o n w i t h F r e q u e n c y o f S e l f I m p e d a n c e " o f S h e a t h i n a C o a x i a l C a b l e . - 6 9 -F I G 7 . 8 V a r i a t i o n w i t h F r e q u e n c y o f M u t u a l I m p e d a n c e B e t w e e n - S h e a t h a n d C e n t r a l C o n d u c t o r . -70-7.3Plpe Type C a b l e s The impedance c a l c u l a t i o n f o r t h i s t y p e of c a b l e i s one o f t h e most i m p o r t a n t a s p e c t s of t h i s t h e s i s , b e c a u s e u n t i l now t h e r e a r e no f o r m u l a s a v a i l a b l e w h i c h s i m u l t a n e o u s l y c o n s i d e r s k i n and p r o x i m i t y e f f e c t s i n p i p e t y p e c a b l e s . T h i s a l s o c r e a t e s a p r o b l e m i n c h e c k i n g t h e r e s u l t s b e c a u s e t h e r e i s no f o r m u l a a g a i n s t w h i c h t h e a n s w e r s c a n be c o m p a r e d . The program o u t p u t was t h e r e f o r e c h e c k e d f o r s p e c i a l c a s e s where i t i s p o s s i b l e t o o b t a i n t h e a n s w e r s i n d i f f e r e n t ways. One s u c h s p e c i a l c a s e of t h e p i p e t y p e c a b l e i s t h e c o a x i a l c a b l e , where the v a l u e s o b t a i n e d p r e v i o u s l y a g r e e d w i t h t h o s e o b t a i n e d from the p i p e t y p e c a b l e program. Two d i f f e r e n t p r o g r a m s w e r e w r i t t e n , one w h i c h u s e s c i r c u l a r f i l a m e n t s and t h e o t h e r t h a t w h i c h uses t h e " e l e m e n t a l " t y p e p r e v i o u s l y e x p l a i n e d . S q u a r e f i l a m e n t s were n o t u s e d , because t h i s a p p r o a c h r e q u i r e s an e x c e s s i v e number of f i l a m e n t s . I n c o n t r a s t t o t h e c a s e of. c o a x i a l c a b l e s , where b e t t e r r e s u l t s a r e o b t a i n e d u s i n g t h e " e l e m e n t a l s " , t h e c i r c u l a r f i l a m e n t s p r o d u c e d more a c c u r a t e a n s w e r s i n t h i s c a s e . T h i s i s due to t h e l a r g e i n f l u e n c e w h i c h p r o x i m i t y e f f e c t s have i n t h i s t y p e o f c o n f i g u r a t i o n . To o b t a i n r e s u l t s w i t h e l e m e n t a l f i l a m e n t s , w i t h an a c c u r a c y s i m i l a r t o t h a t o b t a i n e d w i t h c i r c u l a r f i l a m e n t s , t h e a n g l e span of e a ch f i l a m e n t has t o be v e r y s m a l l , which i m p l i e s t h a t many more elements are needed than f o r t h e c a s e o f c i r c u l a r f i l a m e n t s . T h i s i s t h e r e a s o n why i n a l l t h e c a s e s t u d i e s f o r p i p e t y p e c a b l e s t h e c i r c u l a r f i l a m e n t s -71-program was s e l e c t e d . N e v e r t h e l e s s , t h e f a c t t h a t two d i f f e r e n t p r o g r a m s gave e s s e n t i a l l y t h e same r e s u l t s was an a d d i t i o n a l check t h a t the computer program worked p r o p e r l y . 7.3.1 V a r i a t i o n of the [Z] m a t r i x " w i t h p e r m e a b i l i t y o f t h e p i p e . As m e n t i o n e d i n t h e t h e o r e t i c a l p a r t , t h e n o n l i n e a r i t y i s not t a k e n i n t o a c c o u n t d i r e c t l y i n t h i s program. I n s t e a d , the w h o l e s t e e l p i p e i s a s s u m e d t o be l i n e a r w i t h c o n s t a n t p e r m e a b i l i t y . I n o r d e r t o a s s e s s t h e e f f e c t o f s a t u r a t i o n on the i m p e d a n c e m a t r i x , i m p e d a n c e v a l u e s w i t h e a r t h r e t u r n were o b t a i n e d f o r d i f f e r e n t v a l u e s o f m a g n e t i c p e r m e a b i l i t y o f t h e p i p e a t a g i v e n f r e q u e n c y , 1000Hz. The r e s u l t s a r e i n the form of a 4x4 m a t r i x ( 3 c o n d u c t o r s + p i p e ) , and as e x p e c t e d , t h e o n l y v a l u e s a f f e c t e d by t h e p e r m e a b i l i t y a r e t h o s e o f t h e p i p e , as shown i n t h e T a b l e 7.8. The p i p e t y p e c a b l e under s t u d y w i t h i t s p h y s i c a l and g e o m e t r i c a l c h a r a c t e r i s t i c s i s t h a t o f F i g u r e 7.9. As can be seen f r o m t h e s e r e s u l t s t h e v a r i a t i o n i n t h e r e l a t i v e p e r m e a b i l i t y f r o m 980 t o 100 c h a n g e s t h e r e a c t a n c e ( c o n s i d e r i n g e a r t h r e t u r n ) o n l y by 18%. I t i s t h e r e f o r e p o s s i b l e t o say t h a t f o r t h e c a s e o f p i p e s made o f n o n - l i n e a r m a g n e t i c m a t e r i a l s u c h a s s t e e l , t h e e r r o r made i n n o t c o n s i d e r i n g s a t u r a t i o n i s f a i r l y s m a l l ( l e s s than 18% i n t h i s c a s e ) . R = 0.02425 m. R ±= 0.1095 m. R Q= 0.1159 m. R 1 = R 2 = 0.06440 m. R 3= 0.043825 m. p c o n d u c t o r = 0.293599 E-07 ft p s h e a t h = 0.255905 E-03 Qm F i g . 7.9 P i p e Type Cable -73-TABLE 7.8 Change i n P i p e Parameters w i t h P e r m e a b i l i t y of the P i p e R e l a t i v e P e r m e a b i l i t y P i p e Reactance ( f t / k m ) 980 14.013 920 13.838 650 13.053 200 11.744 100 11.453 I t s h o u l d be r e c a l l e d t h a t as f a r as a.c. i s c o n c e r n e d , t h e m a g n e t i c p e r m e a b i l i t y o f a n o n l i n e a r m a t e r i a l i s not c o n s t a n t t h r o u g h o u t the c y c l e , but changes w i t h the i n s t a n t a n e o u s v a l u e of c u r r e n t . By a s s u m i n g c o n s t a n t p e r m e a b i l i t y f o r t h e s t e e l p i p e , an i n h e r e n t e r r o r i s b e i n g made. Good r e s u l t s w i l l be o b t a i n e d o n l y i f t h e assumed p e r m e a b i l i t y a p p r o x i m a t e s t h e a c t u a l v a l u e f o r most of t h e range t h r o u g h w h i c h t h e c u r r e n t o r f l u x d e n s i t y i n t h e p i p e v a r i e s . T h i s e r r o r c a n o n l y be a v o i d e d i f p h a s o r s o l u t i o n s a r e r e p l a c e d by s o l u t i o n s u s i n g i n s t a n t a n e o u s v a l u e s as a f u n c t i o n o f t i m e . I n t h a t c a s e , r e s i s t a n c e s and i n d u c t a n c e s would depend on t h e c u r r e n t . 7.3.2 V a r i a t i o n of t h e [Z] m a t r i x w i t h f r e q u e n c y . A n o t h e r p o i n t s t u d i e d was t h e v a r i a t i o n of the [Z] m a t r i x w i t h f r e q u e n c y . T h e c o n d u c t o r s i n t h e p i p e a r e n o t s y m m e t r i c a l l y s i t u a t e d , as seen i n f i g u r e 7.9. T h e r e f o r e , t h e - 7 4 -d i a g o n a l e l e m e n t Z ^ o f t h e m a t r i x [ Z ] ( e q u a t i o n 3 . 1 4 ) w i l l b e d i f f e r e n t f r o m Z J J a n d Z 2 2 ' w ^ i ^ e ~ Z 2 2 * * n o r c * e r t o s h o w t h e v a r i a t i o n o f t h e i m p e d a n c e m a t r i x w i t h f r e q u e n c y , t h e v a r i a t i o n o f t h e d i a g o n a l t e r m s Z j ^ . Z ^ . Z ^ i s s u m m a r i z e d i n T a b l e 7 . 9 . T A B L E 7 . 9 F r e q u e n c y V a r i a t i o n o f P i p e T y p e C a b l e M a t r i x F r e q u e n c y ( H z ) z l l ( f t / k m ) Z 3 3 ( f t / k m ) Z 4 4 ( f t / k m ) R X R X R X 0 . 1 0 . 0 1 9 7 7 0 . 0 0 1 9 2 1 0 . 0 1 9 7 7 4 0 . 0 0 1 9 2 1 7 3 . 6 5 7 0 . 0 0 1 8 8 1 . 0 0 . 0 2 0 6 6 0 . 0 1 7 7 6 4 0 . 0 2 0 6 6 6 0 . 0 1 7 7 6 4 7 3 . 6 5 8 0 . 0 1 7 4 0 1 0 . 0 0 . 0 2 9 9 3 0 . 1 6 3 1 0 0 . 0 2 9 9 0 9 0 . 1 6 3 1 1 7 3 . 6 6 7 0 . 1 5 9 5 3 1 0 0 . 0 0 . 1 3 1 5 9 1 . 4 0 7 6 0 . 1 3 1 0 4 1 . 4 7 1 4 7 3 . 7 5 6 1 . 4 5 0 5 1 0 0 0 . 0 1 . 0 4 8 7 0 1 3 . 1 2 3 0 1 . 0 4 6 6 0 1 3 . 1 3 6 7 4 . 6 5 2 1 1 . 1 6 5 1 0 0 0 0 . 0 1 0 . 4 2 9 1 1 6 . 4 5 1 0 . 2 8 6 1 1 6 . 6 2 8 3 . 7 7 6 1 1 5 . 8 9 1 0 0 0 0 0 . 0 1 0 8 . 9 0 1 0 0 7 . 2 1 0 . 7 9 0 1 0 1 3 . 1 1 8 0 . 2 5 1 0 0 8 . 8 6 I n t h i s c a s e t h e p i p e w a s s i m u l a t e d w i t h o n l y o n e l a y e r o f s u b c o n d u c t o r s . T h i s m e a n s t h a t t h e i n c r e a s e o f r e s i s t a n c e w i t h f r e q u e n c y i n t h e p i p e i s d u e t o t h e i n c r e a s e i n s k i n e f f e c t i n t h e c o n d u c t o r s . T h e c u r r e n t s t h r o u g h t h e c o n d u c t o r s a r e f l o w i n g m a i n l y i n t h e o u t e r p a r t o f t h e c o n d u c t o r s a n d t h i s i n c r e a s e s t h e p r o x i m i t y e f f e c t o n t h e p i p e . A s c a n b e s e e n , p r o x i m i t y e f f e c t i s t a k e n i n t o a c c o u n t i n t h i s m e t h o d b e c a u s e Z V 1 . ^ Z , 0 . I t i s o b s e r v e d t h a t R - ^ R , . , : -75-a n d Z 3 3 > Z 1 1 a s i t s h o u l d be , b e c a u s e c o n d u c t o r 1 i s c l o s e r t o the p i p e than c o n d u c t o r 3. 7.3.3 P i p e Type Cable C o n f i g u r a t i o n s T h r e e d i f f e r e n t p i p e t y p e c a b l e c o n f i g u r a t i o n s were s t u d i e d t o s e e w h i c h o n e h a d t h e h i g h e s t l o s s e s . T h e c o n f i g u r a t i o n s were (see F i g u r e s 7.10-7.12): a) C r a d l e b) C l o s e T r i a n g u l a r c) 0pen T r i a n g u l a r I t i s w e l l known [14] [ 2 1 ] , t h a t the c r a d l e c o n f i g u r a t i o n h a s a l a r g e r R a c / R d c r a t i o t h a n t h e c l o s e t r i a n g u l a r c o n f i g u r a t i o n . The [ Z ] m a t r i x f o r t h e c a b l e s o f F i g u r e s 7.10 and 7.11 a t a f r e q u e n c y of 60Hz were o b t a i n e d as .1013+J.8802 .0617+J.8449 .0970+J.8957 symmetric ft/km [Z] = c r a d l e .0617+J.8449 .0521+J.8164 .096+J.8957 .0593+J.7769 .0593+J.7769 .059+J.7769 73.716+j.8995 .0954+J.8936 .0597+J.8449 .0954+j.8936 symmetric ft/km [Z] = c l o s e d t r i a n -g u l a r .0588+J.8369 .0598+J.8369 .0957+J.8933 .0594+J.7769 .0593+J.7769 .0594+J.7769 73.716+j.898 C o m p a r i n g b o t h m a t r i c e s i t i s p o s s i b l e t o s e e t h a t -76-R-^j ( c r a d l e ) > R-j^ ( c l o s e t r i a n g u l a r ) The c u r r e n t c a r r y i n g c a p a c i t y o f t h e c l o s e t r i a n g u l a r c o n f i g u r a t i o n i s t h e r e f o r e h i g h e r t h a n t h a t o f t h e c r a d l e c o n f i g u r a t i o n . F o r t h e open t r i a n g u l a r c o n f i g u r a t i o n , t h e l o s s e s a r e i n f l u e n c e d by p r o x i m i t y e f f e c t s between t h e c o n d u c t o r s and by p r o x i m i t y e f f e c t s b e t w e e n c o n d u c t o r a n d p i p e . I n t h i s c o n f i g u r a t i o n , t h e d i s t a n c e between c o n d u c t o r s i s l a r g e w h i l e the d i s t a n c e between c o n d u c t o r and p i p e i s s m a l l . Due t o t h e h i g h r e s i s t i v i t y o f t h e p i p e , t h e c u r r e n t t h r o u g h i t s h o u l d be s m a l l . The p r o x i m i t y e f f e c t between c o n d u c t o r and p i p e s h o u l d t h e r e f o r e be l e s s i m p o r t a n t than the one between c o n d u c t o r s . As a c o n s e q u e n c e , t h i s c o n f i g u r a t i o n s h o u l d have t h e l o w e s t l o s s e s . The [Z] m a t r i x f o r t h i s case i s : [Z] = open t r i a n -g u l a r 0843+j.9078 ,0596+j.7513 .0847+J.9073 symmetric ,0596+j.7512 .0597+J.7709 .0847+j.9073 0593+J.7769 .0593+J.7769 .0593+J.7769 73.716+J.899 ft/km F i g . 7.10 C r a d l e c o n f i g u r a t i o n F i g . 7.11 C l o s e t r i a n g u l a r c o n f i g u r a t i o n F i g . 7.12 Open t r i a n g u l a r c o n f i g u r a t i o n -78-7.3.4 C u r r e n t D i s t r i b u t i o n i n P i p e Type C a b l e s The l a s t c a s e s t u d y f o r p i p e t y p e c a b l e s i s c o n c e r n e d w i t h t h e c u r r e n t d i s t r i b u t i o n i n s i d e t h e s u b c o n d u c t o r s _ w h i c h make up the p i p e . For t h i s c a s e , the p i p e type system was m o d e l l e d as shown i n F i g u r e 7.13. E a c h f i l a m e n t o f t h e p i p e i s d e f i n e d by i t s a n g u l a r p o s i t i o n w i t h r e s p e c t to a z e r o r e f e r e n c e , and t h o s e of the c o n d u c t o r s a r e d e f i n e d by t h e d i s t a n c e t o t h e c e n t e r and the angle w i t h r e s p e c t t o t h e i r z e r o r e f e r e n c e . The c u r r e n t d i s t r i b u t i o n was o b t a i n e d f o r t h e case where a v o l t a g e was a p p l i e d between one of t h e c o n d u c t o r s and the p i p e . The c u r r e n t w i l l t h e n o n l y f l o w t h r o u g h t h a t c o n d u c t o r and t h e p i p e . No e a r t h r e t u r n was c o n s i d e r e d . The c u r r e n t d i s t r i b u t i o n i n the f i l a m e n t s of the p i p e and the s u b c o n d u c t o r s were o b t a i n e d f o r the f o l l o w i n g c a s e s : a) Only one c a b l e , l o c a t e d f n the c e n t e r of the p i p e ( F i g u r e 7.14). b) Only one c a b l e , l o c a t e d i n an e c c e n t r i c p o s i t i o n ( F i g u r e 7.15). c) Three c a b l e s , but c u r r e n t f l o w i n g t h rough o n l y one of them ( c o n d u c t o r 1) and t h e p i p e . ( F i g u r e 7.16). I n t h e f i r s t c a s e , o n l y s k i n e f f e c t i s o b s e r v e d because, e v e n t h o u g h t h e r e a r e p r o x i m i t y e f f e c t s b e t w e e n t h e c e n t r a l c o n d u c t o r and t h e p i p e , i t a f f e c t s a l l t h e f i l a m e n t s i n the same F i g . 7.13 S u b d i v i s i o n of P i p e Type C a b l e . - 8 0 -p r o p o r t i o n . I n c a s e b ) , b o t h s k i n and p r o x i m i t y e f f e c t s a r e p r e s e n t . I n c a s e c ) , even t h o u g h i t l o o k s s i m i l a r t o c a s e b ) , t h e r e i s a d i f f e r e n c e because of i n d u c e d v o l t a g e s on c o n d u c t o r s 2 and 3 r e s u l t i n g from c u r r e n t f l o w i n c o n d u c t o r 1. These i n d u c e d v o l t a g e s w i l l a l t e r t h e c u r r e n t d i s t r i b u t i o n i n s i d e c o n d u c t o r 1 and i n t h e p i p e as shown l a t e r . F i g . 7.14- One c o n d u c t o r i n c e n t e r of p i p e . F i g . 7.15 One c o n d u c t o r i n e c c e n t r i c p o s i t i o n . F i g . 7.16 Three c o n d u c t o r s i n p i p e . -81-I n F i g u r e 7.17 t h e c u r r e n t d i s t r i b u t i o n i n t h e f i l a m e n t s o f t h e p i p e f o r c a s e s a) and b) was p l o t t e d . The " x " a x i s c o r r e s p o n d s t o t h e a n g u l a r p o s i t i o n " 0 " o f t h e f i l a m e n t and t h e 11 y" a x i s t o t h e c u r r e n t . As c a n be s e e n , t h e c u r r e n t d i s t r i b u t i o n f o r case a) i s u n i f o r m a l l o ver the p i p e due to the abs e n c e o f p r o x i m i t y e f f e c t s . I n case b) t h e c u r r e n t i s g r e a t e r i n t h e f i l a m e n t s w h i c h a r e c l o s e r t o t h e main c o n d u c t o r , and s m a l l e r i n t h o s e w h i c h a r e 180 d e g r e e s a p a r t . F i g u r e 7.18 shows t h e c u r r e n t d i s t r i b u t i o n i n t h e two l a y e r s o f f i l a m e n t s of t h e main c o n d u c t o r f o r t h e same c a s e s . S k i n e f f e c t i n both c a s e s i s e a s y t o s e e , b e c a u s e t h e i n n e r l a y e r ( l a y e r #1) c a r r i e s l e s s c u r r e n t t h a n l a y e r #2. A s l i g h t p r o x i m i t y e f f e c t i s n o t i c e a b l e as w e l l ; f o r case a) t h e c u r r e n t i n each f i l a m e n t o f the same l a y e r i s e q u a l , w h i l e f o r t h e c u r v e s o f c a s e b) t h o s e f i l a m e n t s near t h e p i p e c a r r y s l i g h t l y more c u r r e n t t h a n t h e r e s t o f t h e f i l a m e n t s of the same l a y e r . I t i s i n t e r e s t i n g t o note t h a t t h e c u r r e n t s i n t h e i n n e r l a y e r f l o w i n o p p o s i t e d i r e c t i o n t o t h o s e o f the o u t e r one. These i n d i c a t e s t h a t t h e y a r e c i r c u l a t i n g c u r r e n t s w h i c h i n c r e a s e t h e l o s s e s i n t h e p i p e , a s e x p l a i n e d i n s e c t i o n ( 5 ) . T h i s i s not shown i n t h e c u r v e s , where o n l y a b s o l u t e v a l u e s a r e p l o t t e d . I n F i g u r e 7.19 t h e c u r r e n t d i s t r i b u t i o n i n t h e p i p e i s shown f o r c a s e s b and c . Even though t h e c u r v e s l o o k s i m i l a r t o t h o s e o f F i g u r e 7.17, t h e c u r r e n t i n t h i s c a s e s w i t h t h r e e c o n d u c t o r s i s h i g h e r , w i t h a more u n i f o r m d i s t r i b u t i o n i n t h e f i l a m e n t s o f t h e p i p e . W i t h r e s p e c t t o t h e c u r r e n t d i s t r i b u t i o n i n c o n d u c t o r #1, as shown i n F i g u r e 7.20, t h e i n f l u e n c e of the -82-p r o x i m i t y e f f e c t s from t h e c o n d u c t o r s 2 and 3 i s h i g h e r than the e f f e c t o f t h e p i p e , b e c a u s e t h e f i l a m e n t s n e a r t h e c o n d u c t o r s c a r r y more c u r r e n t than t h e ones near t h e p i p e . T h i s happens i n b o t h l a y e r s of f i l a m e n t s . T h i s i s t h e same c o n c l u s i o n as f o r the open t r i a n g u l a r c o n f i g u r a t i o n a n a l y z e d i n t h e p r e c e d i n g s e c t i o n . I t i s i n t e r e s t i n g t o see how t h i s c u r r e n t d i s t r i b u t i o n m o d i f i e s t h e e f f e c t i v e r e s i s t a n c e o f t h e s y s t e m . The v a l u e s of t h e i m p e d a n c e s f o r t h e t h r e e c a s e s a r e shown i n T a b l e 7.10 TABLE 7.10 V a r i a t i o n of Impedance w i t h C o n f i g u r a t i o n of I n t e r n a l Conductors C A S E Impedance ( ft /km) a 73.699 + j 2.0643 b 73.703 + j 2.0643 c 73.711 + j 1.9371 The r e s i s t a n c e f o r t h e c a s e w i t h one c o n d u c t o r i n t h e c e n t e r o f t h e p i p e i s t h e s m a l l e s t o n e , due t o t h e u n i f o r m c u r r e n t d i s t r i b u t i o n i n t h e f i l a m e n t s of t h e p i p e and l a y e r s of t h e main c o n d u c t o r . The l a r g e s t v a l u e of r e s i s t a n c e i s f o r case c ) , w h i c h i s t o be e x p e c t e d b e c a u s e e v e n t h o u g h t h e c u r r e n t d i s t r i b u t i o n i n t h e p i p e i s more u n i f o r m t h a n i n c a s e b) j u s t t h e o p p o s i t e i s t r u e i n t h e c o n d u c t o r . When t h e n e t c u r r e n t t h r o u g h t h e c o n d u c t o r was e v a l u a t e d f o r b o t h c a s e s , i t was s m a l l e r f o r c a s e c ) , w h i c h a l s o i m p l i e s a l a r g e r i m p e d a n c e v a l u e f o r t h i s c o n f i g u r a t i o n . -83-O . O I S UJ T. a. §D.012 A or x z UJ or 0.011 108 0 144.0 180.0 216.0 252.0 288.0 ANGULRR POSITION OF FILAMENT (DEGREES) 324.0 360.0 FIG 7.17 C u r r e n t D i s t r i b u t i o n i n the P i p e . Cases A ) , B ) . or a . 14 0.13 i 0.12 0.11 -0.10-3 0.09 H 0.08 -J § 0.07 o % 0.06 £ 005-3 cc 3 0 04 0.03 0.02 0.01 -0.00 36.0 "72.0 Layer 2 — CASE A + CASE B Layer 1 10B 0 144.0 180.0 216.0 252.0 288.0 ANGULAR POSITION OF FILAMENT (DEGREES) 324.0 360.0 FIG 7.18 C u r r e n t D i s t r i b u t i o n i n I n t e r n a l Conductor. Cases A ) , B) . -84-FIG 7.19 C u r r e n t D i s t r i b u t i o n i n P i p e . Cases B ) , C ) . 0, 14 n 0.13-0.12 -0.11 0 . 1 0 - 3 0.09 0.08 -3 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 Layer 2 — CASE B + CASE C Layer 1 ' * * ' 1 1 1 1 ' ' 1 • • i ' 1 • 1 i 1 • 1 1 i 1 1 • • i 1 • 1 • i • • • • i • * * • i • .• ' ' i 0 36.0 72.0 108.0 144.0 180.0 216.0 252.0 288.0 324.0 360.0 RNGULflR POSITION OF FILAMENT (OCTREES)• FIG 7.20 C u r r e n t D i s t r i b u t i o n i n I n t e r n a l Conductor. Cases B ) , C).. -85-7.4 Stranded Conductors In t h i s s e c t i o n G a l l o w a y ' s f o r m u l a Rc = Xc = K • ( toy p ) ft /m / T r T (2+n) w i t h K=2.25 w i l l be c o m p a r e d w i t h t h e s u b d i v i s i o n m e t h o d . T h i s f o r m u l a , a c c o r d i n g t o t h e a u t h o r s [ 2 0 ] , g i v e s a c c e p t a b l e r e s u l t s f o r f r e q u e n c i e s above 2-5 kHz. As a t e s t c a s e , the s t r a n d e d c o n d u c t o r of F i g u r e 7.21 was used. The i m p e d a n c e s were c a l c u l a t e d a t s e v e r a l f r e q u e n c i e s . The r e s u l t s f o r t h e r e s i s t a n c e v a l u e s a r e s u m m a r i z e d i n T a b l e 7.11. As can be s e e n , t h e r e s u l t s o b t a i n e d f r o m t h e f o r m u l a and from the program a r e d i f f e r e n t . TABLE 7.11 V a r i a t i o n w i t h Frequency of the R e s i s t a n c e  of a One L a y e r S t r a n d e d Conductor Frequency (Hz) R e s i s t a n c e (Program) ft/km R e s i s t a n c e (Formula)ft/km 20000 1.7122 1.7335 43000 1.8466 2.5418 80000 2.4815 3.4669 100000 2.8160 3.8762 130000 3.2965 4.4195 -86-The f a c t o r K, w h i c h i n G a l l o w a y ' s f o r m u l a i s 2.25 was measured by f i e l d p l o t t i n g i n an e l e c t r o l y t i c t a n k . The r e s u l t s of T a b l e 7.11 were o b t a i n e d by i n c r e m e n t i n g a t a g i v e n f r e q u e n c y t h e number o f f i l a m e n t s u n t i l t h e v a l u e o f R t e n d e d t o a c c o n s t a n t v a l u e . For t h e r e s u l t s of T a b l e 7.11 a v a l u e f o r K o f 1.6395 was o b t a i n e d w i t h t h e s u b d i v i s i o n method e x c e p t a t 20 kHz where t h e f a c t o r was 2.22. The r e s i s t a n c e was a l s o c a l c u l a t e d f o r t h e c a s e o f 19 s t r a n d s ( two l a y e r s of s t r a n d s ) and the r e s u l t s were: TABLE 7.12 V a r i a t i o n w i t h Frequency of the R e s i s t a n c e  of a Two Layer Stranded Conductor Frequency Hz R e s i s t a n c e (Program) ft/km R e s i s t a n c e (Formula) ft/km 43000 1.3479 1.6945 80000 1.6305 2.3113 100000 1.8187 2.5841 130000 2.1126 2.9463 Once a g a i n t h e f a c t o r was f o u n d t o be n e a r t o o f 1.6 than 2.25. The d i f f e r e n t r e s u l t s o b t a i n e d w i t h t h e s e two methods cannot be e a s i l y j u s t i f i e d . From a t h e o r e t i c a l p o i n t o f v i e w , the method o f s u b d i v i s i o n i s more a c c u r a t e , b u t G a l l o w a y ' s f o r m u l a matches f i e l d t e s t e s p e c i a l l y a t lower f r e q u e n c i e s [ 2 0 ] , -87-S . C r i s t i n a a n d M . D ' A m o r e [ 2 9 ] f a c e d t h e s a m e p r o b l e m w h e n c o m p a r i n g t h e i r f o r m u l a w i t h e q u a t i o n 6 . 1 a n d t h e y g a v e p r e f e r e n c e t o G a l l o w a y ' s f o r m u l a , b e c a u s e o f " t h e a c c u r a c y s h o w n a s c o m p a r e d w i t h t h e e x p e r i m e n t a l v a l u e s a n d b e c a u s e o f t h e s i m p l i c i t y o f t h e c a l c u l a t i o n " . T h i s d i f f e r e n c e r e m a i n s t h e r e f o r e a n o p e n i s s u e f o r f u t u r e r e s e a r c h i n t h i s a r e a . p -" 3 . 2 1 1 C T 8 fim U r = 1 n a. 6 F i g . 7 . 2 1 S t r a n d e d C o n d u c t o r . - 8 8 -8 . C O N C L U S I O N S T h e m e t h o d o f s u b d i v i s i o n o f c o n d u c t o r s h a s b e e n u s e d t o c a l c u l a t e t h e i m p e d a n c e o f c a b l e s . T h r e e d i f f e r e n t t y p e s o f f i l a m e n t s w e r e c o n s i d e r e d , a n d f o r a l l t h r e e c a s e s , t h e a c c u r a c y o f t h e c a l c u l a t i o n w a s f o u n d t o d e p e n d o n t h e n u m b e r o f s u b c o n d u c t o r s u s e d . T h e a d v a n t a g e s a n d d i s a d v a n t a g e s o f e a c h t y p e o f f i l a m e n t c a n b e s u m m a r i z e d a s f o l l o w s 1 - S q u a r e t y p e f i l a m e n t s g i v e e x c e l l e n t r e s u l t s , b u t a t h i g h f r e q u e n c i e s ( a b o v e 0 . 1 M H z ) t h i s m e t h o d r e q u i r e s a m u c h l a r g e r n u m b e r o f f i l a m e n t s t h a n t h e o t h e r s u b d i v i s i o n m e t h o d s f o r t h e s a m e d e g r e e o f a c c u r a c y . 2 - E l e m e n t a l t y p e . T h i s i s a v e r y a p p e a l i n g t y p e o f f i l a m e n t , b e c a u s e t h e c r o s s - s e c t i o n a r e a s o f t h e f i l a m e n t s c a n b e l a r g e r t h a n f o r t h e o t h e r t w o t y p e s . T h i s m e a n s t h a t a s m a l l e r n u m b e r o f f i l a m e n t s i s r e q u i r e d f o r t h e s a m e d e g r e e o f a c c u r a c y . H o w e v e r , t h e u s e r o f t h e p r o g r a m m u s t h a v e s o m e p r i o r k n o w l e d g e o f t h e c u r r e n t d i s t r i b u t i o n , t o a v o i d u n a c c e p t a b l e e r r o r w i t h l a r g e a r e a f i l a m e n t s . A n o t h e r d i s a d v a n t a g e o f t h i s m e t h o d i s t h e h i g h c o m p u t e r t i m e w h i c h i s n e e d e d t o o b t a i n t h e i m p e d a n c e s b e t w e e n t w o e l e m e n t a l s . F o r a l l t h e s e r e a s o n s i t i s b e s t t o u s e t h i s a p p r o a c h o n l y f o r s i m p l e c o n f i g u r a t i o n s . 3 - C i r c u l a r f i l a m e n t s . T h i s t y p e o f f i l a m e n t w a s u s e d i n - 8 9 -most of t h e c a s e s d i s c u s s e d i n t h i s t h e s i s . The o n l y problem w i t h i t i s t h a t , f o r c e r t a i n c o n f i g u r a t i o n s , a l a r g e number o f f i l a m e n t s i s needed. A l l t h e f o r m u l a s i n v o l v e d i n t h e p r o c e d u r e a r e v e r y e a s y t o compute, however. A l l t h r e e methods t a k e b o t h s k i n and p r o x i m i t y e f f e c t s i n t o a c c o u n t , as has been shown f o r t h e c u r r e n t d i s t r i b u t i o n i n the p i p e . They a r e t h e r e f o r e w e l l s u i t e d f o r d i s t r i b u t i o n c a b l e s where the c o n d u c t o r s run c l o s e t o g e t h e r . The method e m p l o y e d i n t h i s t h e s i s i s a l s o s u i t e d f o r m o d e l l i n g c o n d u c t o r s w i t h n o n u n i f o r m p r o p e r t i e s a c r o s s t h e s e c t i o n . C a b l e s e n c l o s e d i n m a g n e t i c p i p e s c a n a l s o be c o n s i d e r e d , b u t t h e s a t u r a t i o n i s n o t t a k e n i n t o a c c o u n t d i r e c t l y . A n e x t s t e p t o w a r d s o b t a i n i n g a c o m p l e t e c a b l e p a r a m e t e r s program would be to i n t r o d u c e t h e c a l c u l a t i o n of t h e s h u n t a d m i t t a n c e m a t r i x [ Y ] . The i n c o r p o r a t i o n o f t h i s f e a t u r e i n t o t h e p i p e t y p e c a b l e p r o g r a m s h o u l d n o t be d i f f i c u l t i n t e r m s o f p r o g r a m m i n g , b e c a u s e t h e c a l c u l a t i o n of t h e [Y] m a t r i x i s t o t a l l y d e c o u p l e d f r o m t h e c a l c u l a t i o n o f t h e [Z] m a t r i x . An a s p e c t t h a t r e m a i n u n r e s o l v e d i n c o n n e c t i o n w i t h s t r a n d e d c o n d u c t o r s , i s t h e e f f e c t of s p i r a l l i n g o f t h e s t r a n d s and s h o u l d be the o b j e c t of f u t u r e r e s e a r c h . -90-LIST OF REFERENCE [ I ] P. G r a n e a u , " U n d e r g r o u n d Power T r a n s m i s s i o n " , John W i l e y & Sons" ,1979. [2] E. C o m e l l i n i , A. I n v e r n i z z i , G. Manzoni , "A Computer Program f o r d e t e r m i n i n g E l e c t r i c a l R e s i s t a n c e and R e a c t a n c e o f any T r a n s m i s s i o n L i n e " , IEEE T r a n s . PAS-92 pp. 308-314,1973. [ 3 ] L.M. W e d e p o h l and D . J . W i l c o x , " T r a n s i e n t A n a l y s i s o f U n d e r g r o u n d Power T r a n s m i s s i o n S y s t e m s " , P r o c . I E E , v o l . l 2 0 , No. 2, Feb. 1973. [ 4 ] R. L u c a s and S. T a l u k d a r , " A d v a n c e s i n F i n i t e E l e m e n t T e c h n i q u e s f o r C a l c u l a t i n g C a b l e R e s i s t a n c e s a n d I n d u c t a n c e s " , IEEE T r a n s . PAS-97,No.3, May/June 1978. [5] J.R. C a r s o n , "Ground R e t u r n Impedance, Underground W i r e w i t h E a r t h R e t u r n " , B e l l S y s t . Tech. J o u r , v o l . 8 , 1929, pp.94-98. [6] J.R. C a r s o n , "Wave P r o p a g a t i o n i n Overhead W i r e s w i t h Ground R e t u r n " , B e l l S y s t . Tech. J o u r , v o l . 5,1926, pp. 539-554. [7] J . H. Neher, "Phase Sequence Impedance of P i p e Type C a b l e s " , I E E E T r a n s , on P o w e r A p p a r . & S y s t e m s , A u g u s t 1 9 6 4 , pp.795-804. [ 8 ] E. R. Thomas and R.H. K e r s h a w , " I m p e d a n c e o f P i p e Type C a b l e " , IEEE T r a n s . PAS-84, No. 10, O c t . 1965, pp. 953-961. [9] F. P o l l a c z e k , "Sur l e champ p r o d u i t par un c o n d u c t e u r s i m p l e i n f i n i m e n t l o n g p a r c o u r c e par un c o u r a n t a l t e r n a t i f " , " R e v u e G e n e r a l de L " e l e c t r i c i t e " , Tome X X I X , No.22, May 1 9 3 1 . [ 1 0 ] H.W. Dommel, " E l e c 553- Advanced A n a l y s i s o f Power Systems, C l a s s n o t e s " , U n i v . of B.C.", 1978. [ I I ] " E l e c t r i c a l T r a n s m i s s i o n and D i s t r i b u t i o n R e f e r e n c e Book", F o u r t h E d i t i o n , Westinghouse E l e c t r i c a l Co.,1964. [ 1 2 ] A. M e k j i a n and M. S o s n o w s k i , " C a l c u l a t i o n o f A l t e r n a t i n g C u r r e n t s L o s s e s i n S t e e l P i p e s c o n t a i n i n g Power C a b l e s " , IEEE T r a n s . 82 SM 368-9, 1982. [13] P. Graneau, " A l t e r n a t i n g and T r a n s i e n t Conductor C u r r e n t s i n S t r a i g h t C o n d u c t o r s of any C r o s s - s e c t i o n " , B e l l S y s t . Tech. J o u r . 1965, pp. 40-59. [14] J . P o l t z . E . K u f f e l , "Eddy C u r r e n t L o s s e s i n P i p e t y p e Cable Systems", IEE T r a n s . PAS-101, 1982, pp.825-832. -91-[15] A.E. R u e h l y , "Survey of Com p u t e r - A i d e d E l e c t r i c a l A n a l y s i s o f I n t e g r a t e d C i r c u i t I n t e r c o n e c t i o n " , IBM J o u r n a l of Res e a r c h and Development, v o l . 23, number 61979, pp.626-639. [16] P. S i l v e s t e r , "Modal Network Theory o f S k i n E f f e c t i n F l a t C o n d u c t o r s " , IEEE T r a n s . PAS-54, 1966, pp. 1147-1151. [17] W.T. Weeks, L.L. Wu, M.F.Mc A l l i s t e r , A. S i n g h , " R e s i s t i v e and I n d u c t i v e S k i n E f f e c t i n R e c t a n g u l a r C o n d u c t o r s " , IBM J o u r n a l o f R e s e a r c h and D e v e l o p m e n t , v o l . 23, No.6 , 1979, pp. 652-660. [18] J . Weiss and Z.J. Csendes, "A One Step F i n i t e Element Method f o r M u l t i c o n d u c t o r s S k i n E f f e c t P r o b l e m s " , I E E E T r a n s . 82 WM 102-2. [19] P. S i l v e s t e r , " F i n i t e Element S o l u t i o n of S a t u r a b l e M a g n e t i c F i e l d Problems", IEEE T r a n s . PAS-89 ,1970, pp. 1642-1651. [20] R.H. G a l l o w a y , W.B. S h o r r o c k s and L. W e d e p o h l , " C a l c u l a t i o n o f E l e c t r i c a l P a r a m e t e r s f o r S h o r t and L o n g P o l y p h a s e T r a n s m i s s i o n L i n e s " , P r o c . I E E , v o l . 1 1 1 , No.12 December 1964,pp. 2051-2059. [ 2 1 ] C. E a t z , G.S. E a g e r , G.W. Seman, " P r o g r e s s i n t h e D e t e r m i n a t i o n o f AC/DC R e s i s t a n c e R a t i o s of P i p e Type C a b l e s S y s t e m s " , P r o c . IEEE T r a n s . PAS-97 ,1978, pp. 2262-2269. [ 2 2 ] F. G r o v e r , " I n d u c t a n c e C a l c u l a t i o n s W o r k i n g F o r m u l a s and T a b l e s " , Dover P u b l i c a t i o n s I n c . , New York, 1946. [ 2 3 ] K.O. A b l e d u , " I m p e d a n c e C a l c u l a t i o n o f C a b l e s u s i n g S u b d i v i s i o n s o f t h e C a b l e C o n d u c t o r s " , The U n i v e r s i t y of B r i t i s h Columbia, September 1979. [2 4 ] D.M. Simmons, " C a l c u l a t i o n o f t h e E l e c t r i c a l P r o b l e m s of U n d e r g r o u n d C a b l e s " , The E l e c t r i c J o u r . , v o l . 2 9 , J a n u a r y -December 1932. [ 2 5 ] D.R. S m i t h and J . V . B a r g e r , " I m p edance and C i r c u l a t i o n C u r r e n t C a l c u l a t i o n f o r U.D. M u l t i w i r e C o n c e n t r i c N e u t r a l C i r c u i t s " , I E E E C o n f e r e n c e R e c o r d o f 1971 C o n f e r e n c e on Underground D i s t r i b u t i o n , S e p t . 1971, pp. 992-1000. [ 2 6 ] W.A. L e w i s , G.D. A l l e n and J . C . Wang, "The E l e c t r i c a l C h a r a c t e r i s t i c s o f U n d e r g r o u n d D i s t r i b u t i o n C o n c e n t r i c N e u t r a l C a b l e s " , A . B . C h a n c e C o m p a n y , C e n t r a l i a , M i s s o u r i , 1 9 7 6 [ 27] M.S. Sarma, "S y n c h r o n o u s M a c h i n e s ( T h e i r T h e o r y , S t a b i l i t y a n d E x c i t a t i o n S y s t e m s ) " , G o r d o n a n d B r e a c h , S c i e n c e P u b l i s h e r s I n c . , New York, 1979. -92-[28] S.A. S c h e l k u n o f f , "The E l e c t r o m a g n e t i c Theory of C o a x i a l T r a n s m i s s i o n L i n e and C y l i n d r i c a l S h e l l s " , B e l l System Tech. J o u r . , v o l . 13, pp.522-579, 1934. [29] S. C r i s t i n a and M. D'Amore, " P r o p a g a t i o n on P o l y p h a s e Lossy Power L i n e s : A new Parameter S e n s i t i v i t y M o d el", IEEE T r a n s . PAS-98 pp. 35-44, Jan/Feb 1978. -93-APPENDIX A . F i n i t e Elements and F i n i t e D i f f e r e n c e s A g r e a t amount o f work has been done i n t h e development of t h e s e t e c h n i q u e s d u r i n g t h e l a s t t e n y e a r s . In o r d e r t o show why t h e s e p r o c e d u r e s were not s e l e c t e d f o r t h i s t h e s i s , a b r i e f e x p l a n a t i o n of t h e s e methods i s g i v e n below. A.1 F i n i t e D i f f e r e n c e S o l u t i o n C o n s i d e r a t w o - d i m e n s i o n a l s i m p l y c onnected r e g i o n i n the x,y p l a n e bounded by c o n d u c t o r C. I f t h e c u r r e n t d e n s i t y v e c t o r has a component o n l y i n t h e z d i r e c t i o n o f m a g n i t u d e " J " , t h e n t h e m a g n e t i c v e c t o r p o t e n t i a l A ( x , y ) = A z ( x , y ) s a t i s f i e s t h e n o n l i n e a r P o i s s o n ' s e q u a t i o n : 3 ( V 3A/ 3x) + 3 ( v 3A/ 3y) = - J ( A . l ) 3x 3"y s u b j e c t t o t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n s i n a s p a c e o f m a g n e t i c r e l u c t i v i t y v , w h i c h i s g e n e r a l l y a f u n c t i o n o f b o t h p o s i t i o n and f i e l d . The f l u x d e n s i t y can be e v a l u a t e d from t h e v e c t o r p o t e n t i a l v a l u e s a s : Bx = 3A/ 3y By = - 3A/ 3x (A.2) The s o l u t i o n o f e q u a t i o n A . l y i e l d s t h e d i s c r e t e v a l u e s o f t h e v e c t o r p o t e n t i a l o f t h e m a g n e t i c f i e l d . The c o n t o u r of -94-c o n s t a n t v a l u e s o f A a r e t h e l i n e s o f m a g n e t i c i n d u c t i o n . Consider F i g u r e a . l . The two-dimensional continuum i s s u b - d i v i d e d by h o r i z o n t a l and v e r t i c a l g r i d l i n e s , but o t h e r d i v i s i o n s (e.g., t r i a n g u l a r or p o l a r ) can be used. — -jll^  III 1 F i g . a . l S u b d i v i s i o n of a two dimensional continuum. A s m a l l area bounded by two a d j a c e n t v e r t i c a l g r i d l i n e s and two a d j a c e n t h o r i z o n t a l g r i d l i n e s i s denoted as a mesh. Un i f o r m square meshes over the whole c r o s s - s e c t i o n would o f f e r g r e a t c o m p u t a t i o n a l advantages but such a scheme i s not e n t i r e l y p o s s i b l e . In o r d e r to s a t i s f y boundary c o n d i t i o n s , a v e r y f i n e g r a t i n g o f t h e g r i d would have t o be c h o s e n a t and n e a r the b o u n d a r i e s , so the i d e a i s to choose a f i n e r mesh where g r e a t accuracy i s r e q u i r e d to s a t i s f y boundary c o n d i t i o n s and a c o a r s e r mesh i n t h e i n t e r i o r o f t h e r e g i o n s w h e r e t h e c h a n g e i n r e l u c t i v i t y i s not too l a r g e . The r e l u c t i v i t y may be d e f i n e d a t t h e c e n t e r of each mesh, as being common to the area of the mesh. In order to a v o i d a s e p a r a t e l o c a t i o n i n the computer program f o r the r e l u c t i v i t y , the r e l u c t i v i t y of each mesh may be c o n v e n i e n t l y a s s i g n e d to one of the c o r n e r s o f the mesh c o n s i s t e n t l y . The c u r r e n t d e n s i t y may a l s o be s p e c i f i e d i n a s i m i l a r manner. As,an example r e f e r r i n g to -95-F i g u r e a . 2 , t h e x and y components o f t h e f l u x d e n s i t y a t t h e c e n t e r of the r e c t a n g l e 0 1 8 4 are c a l c u l a t e d a s : 4 I > •> (. SJ 3> } TTI| •I' t 7 + 1 i T 1> *"4 B 1 1 M » 1—hj—1 F i g . a.2 T y p i c a l g r i d p o i n t 0 , i t s s u r r o u n d i n g meshes and g r i d p o i n t s . B x = A R Ag + A Q -A 4 B = - ( A ^ - A Q+ A 8 - A 4 ) 2 h, where A Q , A J , A Q , A 4 are the v e c t o r p o t e n t i a l s a t the g r i d p o i n t s 0 , 1 , 8 , 4 r e s p e c t i v e l y . The a b s o l u t e v a l u e c a n t h e n be c o m p u t e d and be u s e d f o r t h e e v a l u a t i o n o f t h e a p p r o p r i a t e r e l u c t i v i t y i n t h e mesh 0 1 8 4 . The p a r t i a l d i f f e r e n t i a l e q u a t i o n i s n e x t t r a n s f o r m e d i n t o a d i f f e r e n c e e q u a t i o n t o o b t a i n n u m e r i c a l s o l u t i o n s . I n o r d e r t o f i n d t h e f i n i t e d i f f e r e n c e e x p r e s s i o n f o r t h e v e c t o r p o t e n t i a l A Q a t g r i d p o i n t 0 i n t e r m s o f t h e v e c t o r p o t e n t i a l A j , , A j , a n d A^' a t t h e g r i d p o i n t s 1 , 2 , 3 a n d 4 t h e p a r t i a l d e r i v a t i v e s i n e q u a t i o n A l a r e r e p l a c e d by f i n i t e p a r t i a l d i f f e r e n c e q u o t i e n t s w i t h t h e use o f the f i r s t two terms of T a y l o r ' s s e r i e s e x p a n s i o n o f v e c t o r p o t e n t i a l A around p o i n t 0 . However, t h e d i f f e r e n c e e x p r e s s i o n f o r t h e v e c t o r p o t e n t i a l a t a t y p i c a l g r i d p o i n t 0 may a l s o be o b t a i n e d by t h e a p p l i c a t i o n - 9 6 -o f Ampere's Law a r o u n d t h e c l o s e d r e c t a n g u l a r c o n t o u r ABCDA shown I n F i g u r e a.2 p a s s i n g t h r o u g h t h e c e n t e r p a r t o f t h e s u r r o u n d i n g meshes. r^ >H d l = 1 ( I = c u r r e n t e n c l o s e d by the c l o s e d c o n t o u r C) | V a b B y I d y + K c B x I I d x + | V c d B y I I I d y + f V d a B X I V ab DC cd da dx= Io Using e q u a t i o n A.2 B y I ~ A o " A i B x I I " A 2 " A 0 h l h 2 B y I I I ~ A 3 " A 0 B x I V " A 0 " A 4 h 3 h 4 | dy may be r e p l a c e d by V 1 ^ 4 + V 2 ^ 2 ab 2 I V ^ dx may be r e p l a c e d by - ( V 2 n j + V 3 n 3 ^ c 2 | V c d d y m a y ^ e r e P ^ - a c e ^ by ~ ( V 3 ^ 2 + V 4 n 4 ^ cd 2 v ^ a dx may be r e p l a c e d by h^ + v ^ h ^ (A.3) da 2 I n s e r t i n g t h e s e e x p r e s s i o n s i n t o e q u a t i o n A3 g i v e s : -97-A = I + ? a, A. . o o T i 11 (A.4) I o = 0.25 [ J1 h 4 h l + J 2 h 1 h 2 + J 3 h 2 h 3 + J 4 h 3 ] ° 1 - V l h 4 + V 2 h a 2 = v 2 h x + v 3 h 3 2 h. 2 h. « 3 - V 3 h 2 + V 4 h 4 % " V 4 h 3 + V l h l 2 h 3 2 h 4 E q u a t i o n A.4 i s u s e d a t a l l g r i d p o i n t s e x c e p t a t t h o s e on t h e o u t e r b o u n d a r i e s , w h e r e t h e D i r i c h l e t b o u n d a r y c o n d i t i o n s t h a t A i s e q u a l t o z e r o i s c h o s e n . T h u s , N s i m u l t a n e o u s e q u a t i o n s w o u l d be o b t a i n e d f o r t h e v e c t o r p o t e n t i a l a t N i n t e r n a l g r i d p o i n t s . An i n i t i a l g u e s s o f t h e v e c t o r p o t e n t i a l s a t a l l g r i d p o i n t s n e e d s t o be made a t t h e b e g i n n i n g o f t h e n u m e r i c a l s o l u t i o n . The s o l u t i o n o f t h e s e t o f N s i m u l t a n e o u s e q u a t i o n s y i e l d s t h e n u m e r i c a l v a l u e s o f t h e v e c t o r p o t e n t i a l s a t t h e N g r i d p o i n t s . F o r t h e c a s e o f c o n s t a n t r e l u c t i v i t y V , e q u a t i o n A . l , when t h e s o u r c e c u r r e n t s a r e a s s u m e d t o be s i n u s o i d a l , t h e e q u a t i o n i s l i n e a r and no i t e r a t i o n s a r e i n v o l v e d . W i t h t h i s m e t h o d i t i s p o s s i b l e t o o b t a i n t h e t y p i c a l f l u x d i s t r i b u t i o n i n t h e c r o s s s e c t i o n a r e a u n d e r s t u d y . -98-A.2 F i n i t e Element S o l u t i o n An a l t e r n a t e a p p r o a c h i s t o f o r m u l a t e the f i e l d p r o b l e m i n v a r i a t i o n a l t e r m s so t h a t i t i s r e d u c e d t o one of m i n i m i z i n g an e n e r g y f u n c t i o n a l r a t h e r t h a n t h a t of s o l v i n g d i f f e r e n t i a l e q u a t i o n s . I t can be shown [ 1 9 ] t h a t t h e s o l u t i o n o f e q u a t i o n A l i s a l w a y s s u c h t h a t i t m i n i m i z e s t h e e n e r g y f u n c t i o n a l F g i v e n by: B F = f f ( f \ db) dx dy - f f J A dx dy JJ J0 b J (A.5) R R where R i s t h e a r e a of d e f i n i t i o n o f t h e p r o b l e m . E q u a t i o n A . l i s t h e E u l e r e q u a t i o n o f t h e f u n c t i o n a l d e f i n e d by e q u a t i o n A.5 s a t i s f y i n g homogeneous D i r i c h l e t and Neumann t y p e o f b o u n d a r y c o n d i t i o n s . The e s s e n c e o f any v a r i a t i o n a l method i s t o s e a r c h d i r e c t l y f o r a f u n c t i o n A w h i c h m i n i m i z e s F. In o r d e r to s e a r c h e f f i c i e n t l y , h o w e v e r , i t i s f i r s t n e c e s s a r y t o d i s c r e t i z e t h e p r o b l e m . The e n t i r e p r o b l e m r e g i o n i s d i v i d e d up i n t o t r i a n g l e s ( f i r s t o r d e r f i n i t e e l e m e n t s ) i n any c o n v e n i e n t f a s h i o n , e n s u r i n g t h a t t h e i n t e r f a c e s o f d i f f e r e n t m a t e r i a l s c o i n c i d e w i t h t r i a n g u l a r s i d e s . An a p p r o x i m a t i o n t o t h e v e c t o r p o t e n t i a l s o l u t i o n A i s assumed i n each t r i a n g l e , such t h a t i t s v a l u e i s a l i n e a r i n t e r p o l a t e o f i t s v a l u e a t t h e t r i a n g u l a r v e r t i c e s . That i s c o n c i s e l y e x p r e s s e d a s : A(x,y) = 1 I ( p i + q ± x + r ± y) A ± 2A (A.6) where the i n d e x i ranges over the t r i a n g l e v e r t i c e s k,m,n and the f o l l o w i n g r e l a t i o n s h o l d : -99-p, =x y - y x q, = y - y r,= x - x *K n r n Jm n •'m J n k m n (A. 7) A = T r i a n g u l a r Area A.,A ,A a r e the v e r t e x v a l u e s of A k' m' m On d i f f e r e n t i a t i n g e q u a t i o n A.6 one o b t a i n s the v a l u e s of f l u x d e n s i t y w i t h i n the t r i a n g l e as B - 1 i ( K q k A k + s r k \ > 2 A (A.8) A A where i and i a r e the u n i t v e c t o r s i n the x and y x y J d i r e c t i o n s r e s p e c t i v e l y . The f l u x d e n s i t y i s assumed t o be a c o n s t a n t w i t h i n each t r i a n g l e . The e x p r e s s i o n s g i v e n above a r e v a l i d j u s t f o r one t r i a n g l e so i t i s n e c e s s a r y t o w r i t e e q u a t i o n s bf t h i s form f o r each t r i a n g l e i n t u r n . M i n i m i z a t i o n o f t h e e n e r g y f u n c t i o n a l i s a c h i e v e d by s e t t i n g i t s f i r s t d e r i v a t i v e w i t h r e s p e c t to each of t h e v e r t e x v a l u e s of t h e p o t e n t i a l t o z e r o , so t h a t f o r a l l k, 3F = 0 9 A k U s i n g e q u a t i o n (A5) i t i s easy t o prove [19] 3F 3A, k '( VB 3B - 3A) dx dy (A.9) 3A, 3A, k I A U n l e s s k i s one o f t h e v e r t i c e s o f t h e t r i a n g l e b e i n g c o n s i d e r e d , t h e d i f f e r e n t i a t i o n w i t h r e s p e c t t o A^ p r o d u c e s a z e r o . So e q u a t i o n A.9 i s , t h e r e f o r e , an e x p r e s s i o n i n t h r e e v a r i a b l e s o n l y . S u b s t i t u t i n g f o r B f r o m e q u a t i o n A.8 and a s s u m i n g J t o be c o n s t a n t t h r o u g h o u t t h e t r i a n g l e , one o b t a i n s -100-V < i < qk qm + r k rm> A k > " A J 4 3 (A.10) When e q u a t i o n s s i m i l a r t o A.10 are w r i t t e n f o r each t r i a n g l e " i n the r e g i o n of i n t e g r a t i o n and t h e c o r r e s p o n d i n g terms a d d e d , one can o b t a i n a s i n g l e m a t r i x e q u a t i o n f o r t h e e n t i r e r e g i o n of the form: [ S ] [ A ] = [A J ] 3 w h e r e [ S ] i s t h e c o m b i n e d c o e f f i c i e n t m a t r i x , [ A ] i s t h e c o l u m n v e c t o r o f v e c t o r p o t e n t i a l s , and [ J ] i s t h e v e c t o r o f c u r r e n t d e n s i t i e s . T h i s i s a s y s t e m o f e q u a t i o n s , w h i c h i n the c a s e o f n o n l i n e a r s y s t e m s may be s o l v e d u s i n g an i t e r a t i v e p r o c e d u r e . A f t e r t h e s o l u t i o n h a s b e e n f o u n d , t h e i n d u c t a n c e c a l c u l a t i o n s a r e done u s i n g the f o r m u l a [27] L = 1 I 2 ' ( A I ) dv F o r c a s e s where t h e c u r r e n t d e n s i t y i s u n i f o r m and t h e model i s o n l y t w o - d i m e n s i o n a l , f u r t h e r s i m p l i f i c a t i o n s e x i s t . A.3 Comparison w i t h S u b d i v i s i o n Method A f t e r t h e d e s c r i p t i o n o f b o t h methods i t i s easy t o see why t h e f i n i t e d i f f e r e n c e o r f i n i t e e l e m e n t t e c h n i q u e was not s e l e c t e d . C o m p a r i n g them w i t h t h e s u b d i v i s i o n method, b o t h use a s u b d i v i d i s i o n o f t h e e n t i r e s y s t e m . I n t h e c a s e o f t h e - 1 0 1 -s u b c o n d u c t o r method, o n l y t h e c o n d u c t i v e p a r t of t h e s t r u c t u r e i s s p l i t i n t o f i l a m e n t s , w h i l e w i t h f i n i t e e l e m e n t t e c h n i q u e s the whole a r e a under s t u d y i s s u b d i v i d e d . T h i s i m p l i e s many more n o d e s o r p o i n t s , and b e c a u s e o f t h i s , more memory s p a c e i s r e q u i r e d ( e v e n t h o u g h nowadays memory space i s not as i m p o r t a n t anymore) . I n g e n e r a l , t h e a l g o r i t h m s f o r f i n i t e e lement t e c h n i q u e s a r e much more c o m p l i c a t e d , not o n l y b e c a u s e o f t h e c r e a t i o n of t h e g r i d , b u t a l s o b e c a u s e i t i s n e c e s s a r y t o e v a l u a t e t h e p o t e n t i a l v e c t o r A i n a l l t h e n o d e s , i n o r d e r t o be a b l e t o i n t e g r a t e t h e f l u x and t h e r e b y c a l c u l a t e t h e s e l f and m u t u a l i n d u c t a n c e between c o n d u c t o r s . I n the case of s u b c o n d u c t o r s , much s i m p l e r f o r m u l a s a r e u s e d f o r t h e m u t u a l i n d u c t a n c e s b e tween f i l a m e n t s . I n b o t h m e t h o d s t h e a s s u m p t i o n o f c o n s t a n t c u r r e n t d e n s i t y i n each f i l a m e n t o r mesh i s made. The d i f f e r e n c e between them i s t h a t f o r t h e f i n i t e e l e m e n t method i t i s n e c e s s a r y t o know t h e t o t a l c u r r e n t or t h e c u r r e n t d i s t r i b u t i o n t h r o u g h t h e c o n d u c t o r , w h i l e i n t h e m e t h o d o f s u b d i v i s i o n t h e c u r r e n t through t h e f i l a m e n t s i s an unknown. Because o f t h e s e r e a s o n s , the method of s u b c o n d u c t o r s i s a much s i m p l e r and more d i r e c t method n e v e r t h e l e s s , f i n i t e e l e m e n t t e c h n i q u e s a r e v e r y u s e f u l f o r t h e c a l c u l a t i o n o f m a g n e t i c f i e l d s , e s p e c i a l l y i n t h e c a s e o f t r a n s f o r m e r s and g e n e r a t o r s w i t h s a t u r a t i o n e f f e c t s . 

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